0
Want this question answered?
Be notified when an answer is posted
Add your answer:
Earn +20 pts
Is it always true that between any two zeros of the derivative of any polynomial there is a zero of the polynomial?
No. Consider the polynomial: f(x) = x3 + 4x2 + 4x + 16 then f'(x) = 3x2 + 8x + 4 = (3x + 2)(x + 2) => x = -2/3, -2 are the zeros of f'(x) Using the second derivative: f''(x) = 6x + 8 it can be seen that: f''(-2) = -4 -> x = -2 is a maximum f''(-2/3) = +4 -> x = -2/3 is a minimum But plugging back into the original polynomial: f(-2) = 16 f(-2/3) = 14 22/27 Between the zeros of the first derivative, the slope of the polynomial is negative so that the polynomial is always decreasing in value, but as the polynomial is greater than zero at the zeros of the first derivative, it cannot become zero between them. That is it has no zeros between the zeros of its first derivative f(x) = x3 + 4x2 + 4x + 16 = (x + 4)(x2 + 4) has only 1 zero at x = -4.
Is it always true that for any polynomial px if x is a zero of the derivative then x px is a maximum or minimum value of px?
No. The important decider is the second derivative of the polynomial (the gradient of the gradient of the polynomial) at the zero of the first derivative: If less than zero, then the point is a maximum If more than zero, then the point in a minimum If equal to zero, then the point is a point of inflection. Consider the polynomial f(x) = x3, then f'(x) = 3x2 f'(0) = 0 -> x = 0 could be a maximum, minimum or point of inflection. f''(x) = 6x f''(0) = 0 -> x = 0 is a point of inflection Points of inflection do not necessarily have a zero gradient, unlike maxima and minima which must. Points of inflection are the zeros of the second derivative of the polynomial.
What if the fourth derivative of a polynomial is zero?
There are many things that can be said about a polynomial function if its fourth derivative is zero, but the main thing you can know about this function from this information is that its order is 3 or less. Consider an nth order polynomial with only positive exponents: axn + bxn-1 + ... + cx2 + dx + e As you derive this function, its derivatives will eventually be equal to zero. The number of derivatives that are nonzero before they all become zero can tell you what order the polynomial function was. Consider an example, y = x4. y = x4 y' = 4x3 y'' = 12x2 y''' = 24x y(4) = 24 y(5) = 0 The original polynomial was of order 4, and its derivatives were nonzero up until its fifth derivative. From this, you can generalize to say that any function whose fifth derivative is equal to zero is of order 4 or less. If the function was of higher order than 4, its derivatives would not become zero until later. If the function was of lower order than 4, its fifth derivative would still be zero, but it would not be the first zero-valued derivative. So this experimentation yielded a rule that the first zero-valued derivative is one greater than the order of the polynomial. Your problem states that some polynomial has a fourth derivative that is zero. Our working rule states that this polynomial can be of highest order 3. So, your polynomial can be, at most, of the form: y = ax3 + bx2 + cx + d Letting the constants a through d be any real number (including zero), this general form expresses any polynomial that will satisfy your condition.
What are the 5 properties of polynomials?
A sum of polynomials is a polynomial.A product of polynomials is a polynomial.A composition of two polynomials is a polynomial, which is obtained by substituting a variable of the first polynomial by the second polynomial.The derivative of the polynomial anxn + an-1xn-1 + ... + a2x2 + a1x + a0 is the polynomial nanxn-1 + (n-1)an-1xn-2 + ... + 2a2x + a1. If the set of the coefficients does not contain the integers (for example if the coefficients are integers modulo some prime number p), then kak should be interpreted as the sum of ak with itself, k times. For example, over the integers modulo p, the derivative of the polynomial xp+1 is the polynomial 0.If the division by integers is allowed in the set of coefficients, a primitive or antiderivative of the polynomial anxn + an-1xn-1 + ... + a2x2 + a1x + a0 is anxn+1/(n+1) + an-1xn/n + ... + a2x3/3 + a1x2/2 + a0x +c, where c is an arbitrary constant. Thus x2+1 is a polynomial with integer coefficients whose primitives are not polynomials over the integers. If this polynomial is viewed as a polynomial over the integers modulo 3 it has no primitive at all.
Does The second derivative represent the rate of change of the first derivative?
Yes.
Is it always true that between any two zeros of the derivative of any polynomial there is a zero of the polynomial?
No. Consider the polynomial: f(x) = x3 + 4x2 + 4x + 16 then f'(x) = 3x2 + 8x + 4 = (3x + 2)(x + 2) => x = -2/3, -2 are the zeros of f'(x) Using the second derivative: f''(x) = 6x + 8 it can be seen that: f''(-2) = -4 -> x = -2 is a maximum f''(-2/3) = +4 -> x = -2/3 is a minimum But plugging back into the original polynomial: f(-2) = 16 f(-2/3) = 14 22/27 Between the zeros of the first derivative, the slope of the polynomial is negative so that the polynomial is always decreasing in value, but as the polynomial is greater than zero at the zeros of the first derivative, it cannot become zero between them. That is it has no zeros between the zeros of its first derivative f(x) = x3 + 4x2 + 4x + 16 = (x + 4)(x2 + 4) has only 1 zero at x = -4.
How different polynomial and non polynomial?
Well, "non-polynomial" can be just about anything; presumably you mean a non-polynomial FUNCTION, but there are lots of different types of functions. Polynomials, among other things, have the following properties - assuming you have an expression of the type y = P(x):* The polynomial is defined for any value of "x". * The polynomial makes is continuous; i.e., it doesn't make sudden "jumps". * Similarly, the first derivative, the second derivative, etc., are continuous. A non-polynomial function may not have all of these properties; for example: * A rational function is not defined at any point where the denominator is zero. * The square root function is not defined for negative values. * The first derivative (i.e., the slope) of the absolute value function makes a sudden jump at x = 0. * The function that takes the integer part of any real number makes sudden jumps at all integers.
Is it always true that for any polynomial px if x is a zero of the derivative then x px is a maximum or minimum value of px?
No. The important decider is the second derivative of the polynomial (the gradient of the gradient of the polynomial) at the zero of the first derivative: If less than zero, then the point is a maximum If more than zero, then the point in a minimum If equal to zero, then the point is a point of inflection. Consider the polynomial f(x) = x3, then f'(x) = 3x2 f'(0) = 0 -> x = 0 could be a maximum, minimum or point of inflection. f''(x) = 6x f''(0) = 0 -> x = 0 is a point of inflection Points of inflection do not necessarily have a zero gradient, unlike maxima and minima which must. Points of inflection are the zeros of the second derivative of the polynomial.
What if the fourth derivative of a polynomial is zero?
There are many things that can be said about a polynomial function if its fourth derivative is zero, but the main thing you can know about this function from this information is that its order is 3 or less. Consider an nth order polynomial with only positive exponents: axn + bxn-1 + ... + cx2 + dx + e As you derive this function, its derivatives will eventually be equal to zero. The number of derivatives that are nonzero before they all become zero can tell you what order the polynomial function was. Consider an example, y = x4. y = x4 y' = 4x3 y'' = 12x2 y''' = 24x y(4) = 24 y(5) = 0 The original polynomial was of order 4, and its derivatives were nonzero up until its fifth derivative. From this, you can generalize to say that any function whose fifth derivative is equal to zero is of order 4 or less. If the function was of higher order than 4, its derivatives would not become zero until later. If the function was of lower order than 4, its fifth derivative would still be zero, but it would not be the first zero-valued derivative. So this experimentation yielded a rule that the first zero-valued derivative is one greater than the order of the polynomial. Your problem states that some polynomial has a fourth derivative that is zero. Our working rule states that this polynomial can be of highest order 3. So, your polynomial can be, at most, of the form: y = ax3 + bx2 + cx + d Letting the constants a through d be any real number (including zero), this general form expresses any polynomial that will satisfy your condition.
Polynomials and non polynomials how different?
Well, "non-polynomial" can be just about anything; presumably you mean a non-polynomial FUNCTION, but there are lots of different types of functions. Polynomials, among other things, have the following properties - assuming you have an expression of the type y = P(x):* The polynomial is defined for any value of "x". * The polynomial makes is continuous; i.e., it doesn't make sudden "jumps". * Similarly, the first derivative, the second derivative, etc., are continuous. A non-polynomial function may not have all of these properties; for example: * A rational function is not defined at any point where the denominator is zero. * The square root function is not defined for negative values. * The first derivative (i.e., the slope) of the absolute value function makes a sudden jump at x = 0. * The function that takes the integer part of any real number makes sudden jumps at all integers.
How different polynomials and non polynomials?
Well, "non-polynomial" can be just about anything; presumably you mean a non-polynomial FUNCTION, but there are lots of different types of functions. Polynomials, among other things, have the following properties - assuming you have an expression of the type y = P(x):* The polynomial is defined for any value of "x". * The polynomial makes is continuous; i.e., it doesn't make sudden "jumps". * Similarly, the first derivative, the second derivative, etc., are continuous. A non-polynomial function may not have all of these properties; for example: * A rational function is not defined at any point where the denominator is zero. * The square root function is not defined for negative values. * The first derivative (i.e., the slope) of the absolute value function makes a sudden jump at x = 0. * The function that takes the integer part of any real number makes sudden jumps at all integers.
What was first matrix revolution or matrix reloaded?
The first movie was "The Matrix", the second was "Matrix Reloaded", then "Matrix Revolutions".
How different the polynomials and non polynomials?
Well, "non-polynomial" can be just about anything; presumably you mean a non-polynomial FUNCTION, but there are lots of different types of functions. Polynomials, among other things, have the following properties - assuming you have an expression of the type y = P(x):* The polynomial is defined for any value of "x". * The polynomial makes is continuous; i.e., it doesn't make sudden "jumps". * Similarly, the first derivative, the second derivative, etc., are continuous. A non-polynomial function may not have all of these properties; for example: * A rational function is not defined at any point where the denominator is zero. * The square root function is not defined for negative values. * The first derivative (i.e., the slope) of the absolute value function makes a sudden jump at x = 0. * The function that takes the integer part of any real number makes sudden jumps at all integers.
What are the 5 properties of polynomials?
A sum of polynomials is a polynomial.A product of polynomials is a polynomial.A composition of two polynomials is a polynomial, which is obtained by substituting a variable of the first polynomial by the second polynomial.The derivative of the polynomial anxn + an-1xn-1 + ... + a2x2 + a1x + a0 is the polynomial nanxn-1 + (n-1)an-1xn-2 + ... + 2a2x + a1. If the set of the coefficients does not contain the integers (for example if the coefficients are integers modulo some prime number p), then kak should be interpreted as the sum of ak with itself, k times. For example, over the integers modulo p, the derivative of the polynomial xp+1 is the polynomial 0.If the division by integers is allowed in the set of coefficients, a primitive or antiderivative of the polynomial anxn + an-1xn-1 + ... + a2x2 + a1x + a0 is anxn+1/(n+1) + an-1xn/n + ... + a2x3/3 + a1x2/2 + a0x +c, where c is an arbitrary constant. Thus x2+1 is a polynomial with integer coefficients whose primitives are not polynomials over the integers. If this polynomial is viewed as a polynomial over the integers modulo 3 it has no primitive at all.
In mathematics what is the value of a score?
What's the score?A score is a unit amount, like a dozen. A score is equal to 20. Alternatively, it is another name for the first derivative. The hessian matrix is the matrix of the second derivatives.
How do you find second derivative of a function?
All it means to take the second derivative is to take the derivative of a function twice. For example, say you start with the function y=x2+2x The first derivative would be 2x+2 But when you take the derivative the first derivative you get the second derivative which would be 2
When was matrix first out?
The Toyota Matrix was first sold in 2002.
