Question

$f(x)=-x^{3}+12 x$ (a) Determine whether $f$ is even, odd, or neither. (b) There is a local maximum value of 16 at $2 .$ Determine the local minimum value.

Name: Problem 76, Chapter 3: College Algebra
Uploaded: 2020-05-07T13:07:31-08:00
Duration: 1 min 31 s
Channel: Ankit Gupta
Description: f(x)=-x^3+12 x (a) Determine whether f is even, odd, or neither. (b) There is a local maximum value of 16 at 2 . Determine the local minimum value.

   $f(x)=-x^{3}+12 x$
(a) Determine whether $f$ is even, odd, or neither.
(b) There is a local maximum value of 16 at $2 .$ Determine the local minimum value.

College Algebra

Michael Sullivan 10th Edition

Chapter 3, Problem 76

Instant Answer

Solved by Expert Ankit Gupta

05/07/2020

Step 1

To do this, we replace $x$ with $-x$ in the function and simplify. $f(-x)=-(-x)^{3}+12(-x) = -x^{3}-12x$ Show more…

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$f(x)=-x^{3}+12 x$ (a) Determine whether $f$ is even, odd, or neither. (b) There is a local maximum value of 16 at $2 .$ Determine the local minimum value.

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Key Concepts

Local Extrema and the Role of Derivatives

This concept focuses on identifying the local maximum and minimum values of a function by using derivatives. Critical points, where the derivative is zero or undefined, are examined with tools such as the first or second derivative tests to determine whether these points correspond to local maxima, local minima, or saddle points. This method is particularly useful in analyzing the behavior of polynomial functions, where the continuity and differentiability ensure that standard calculus techniques apply.

Even and Odd Functions

This concept involves determining the symmetry of a function. An even function satisfies f(?x) = f(x) and is symmetric about the y-axis, whereas an odd function satisfies f(?x) = ?f(x) and is symmetric about the origin. Recognizing whether a function is even, odd, or neither can simplify analysis and graphing by revealing inherent symmetrical properties.

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