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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.
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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.
What function always has a derivative of 0?
f(x) = c, where c is constant, has a derivative of zero.
Find the derivative of 2x2?
2 x 2 = 4. 4 is a constant. The derivative of a constant is always 0. Therefore, The derivative of 2 x 2 is zero.
If the 2nd derivative of an equation isn't constant is it still a quadratic relation?
No. A quadratic equation always has a second derivative that is a constant. For example -3x2 + 10x - 2 first derivative -6x + 10 second derivative -6
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.
Which property of polynomial addition says that the sum of two polynomials is always a polynomial?
It is called the property of "closure".
What is the derivative of square root of 2?
the derivative is 0. the derivative of a constant is always 0.
What property of polynomial multiplication says that the product of two polynomials is always a polynomial?
Clouser
