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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 else can I help you with?
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 can i Differentiate between an algebraic expression and polynomial?
A polynomial is always going to be an algebraic expression, but an algebraic expression doesn't always have to be a polynomial. An algebraic expression is an expression with a variable in it, and a polynomial is an expression with multiple terms with variables in it.
Will the product of two polynomials always be a polynomial?
Yes, the product of two polynomials will always be a polynomial. This is because when you multiply two polynomials, you are essentially combining like terms and following the rules of polynomial multiplication, which results in a new polynomial with coefficients that are the products of the corresponding terms in the original polynomials. Therefore, the product of two polynomials will always be a polynomial.
What function always has a derivative of 0?
f(x) = c, where c is constant, has a derivative of zero.
What property of polynomial subtraction says that the difference of two polynomials is always a polynomial?
The property of polynomial subtraction that ensures the difference of two polynomials is always a polynomial is known as closure under subtraction. This property states that if you take any two polynomials, their difference will also yield a polynomial. This is because subtracting polynomials involves combining like terms, which results in a polynomial expression that adheres to the same structure as the original polynomials.
Is it always true that the zeros of the derivative and the zeros of the polynomial always alternate in location along the horizontal axis?
A zero of the derivative will always appear between two zeroes of the polynomial. However, they do not always alternate. Sometimes two or more zeroes of the derivative will occur between two zeroes of a polynomial. This is often seen with quartic or quintic polynomials (polynomials with the highest exponent of 4th or 5th power).
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 between any two zeros of any polynomial there is a zero of the derivative?
Yes.
Will the minimum an maximum temperature always be equidistant from the current temperature?
Not necessarily. The minimum and maximum temperature can vary and may not always be equidistant from the current temperature. The difference between the current temperature and the minimum or maximum temperature depends on various factors such as weather conditions and time of day.
A polynomial function is always continuous?
Yes, a polynomial function is always continuous
Differentiate of polynomial and algebraic expression?
A polynomial is always going to be an algebraic expression, but an algebraic expression doesn't always have to be a polynomial. In another polynomial is a subset of algebraic expression.
Is the product of two polynomials always a polynomial?
Yes. A polynomial multiplying by a polynomial will always have a multi-termed product. Hope this helps!
Can the average speed ever exceed the maximum speed for a trip?
No, the average speed will always be between the minimum and maximum speeds.
How can i Differentiate between an algebraic expression and polynomial?
A polynomial is always going to be an algebraic expression, but an algebraic expression doesn't always have to be a polynomial. An algebraic expression is an expression with a variable in it, and a polynomial is an expression with multiple terms with variables in it.
What is the derivative of square root of 2?
the derivative is 0. the derivative of a constant is always 0.
Which property of polynomial addition says that the sum of two polynomials is always a polynomial?
It is called the property of "closure".
Can the sum of three polynomials again be a polynomial?
The sum of two polynomials is always a polynomial. Therefore, it follows that the sum of more than two polynomials is also a polynomial.
