Chapter 2: Q 71. (page 91)

$g (x) = x^{3} - 27 x$
(a) Determine whether the given function $g$ is even, odd, or neither.
(b) There is a local minimum value of $- 54$ at $3$ . Determine the local maximum value.

Short Answer

Expert verified

Part (a). The given function $g (x)$ is odd.

Part (b). The local maximum value is $54$ at $x = - 3$ .

Step by step solution

Part (a) Step 1. Given information

The given function is $g (x) = x^{3} - 27 x$ and the local miniumum is $- 54$ at $3$ .

Part (a) Step 2. Find whether the given function g is even, odd, or neither.

A function is even, if $g (- x) = g (x)$ .
A function is odd, if $g (- x) = - g (x)$ .

Replace $x$ by $- x$ in $g (x)$ .

$\begin{array}{rcl} g (- x) & = & {(- x)}^{3} - 27 (- x) \\ = & - x^{3} + 27 x \\ = & - (x^{3} - 27 x) \\ = & - g (x) \end{array}$

Since $g (- x) = - g (x)$ , the given function is odd.

Part (b) Step 1. Determine the point of local maximum value.

Since is an odd function, conclude that the graph is symmetric with respect to the origin.

There is local minimum $- 54$ at $x = 3$ , so there is local maximum at $x = - 3$ .

Part (b) Step 2. Determine the local maximum value at the point x=-3.

The function is odd, so $g (- x) = - g (x)$ .

This implies that $g (- 3) = - g (3) \dots (1)$ .

Substitute the value of $g (3) = - 54$ into $(1)$ .

$\begin{array}{rcl} g (- 3) & = & - g (3) \\ = & - (- 54) \\ = & 54 \end{array}$

Hence, the local maximum value is $54$ at $x = - 3$ .

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$g (x) = x^{3} - 27 x$
(a) Determine whether the given function $g$ is even, odd, or neither.
(b) There is a local minimum value of $- 54$ at $3$ . Determine the local maximum value.

Short Answer

Step by step solution

Part (a) Step 1. Given information

Part (a) Step 2. Find whether the given function g is even, odd, or neither.

Part (b) Step 1. Determine the point of local maximum value.

Part (b) Step 2. Determine the local maximum value at the point x=-3.

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g(x)=x3-27x(a) Determine whether the given function gis even, odd, or neither.(b) There is a local minimum value of -54at 3. Determine the local maximum value.

Short Answer

Step by step solution

Part (a) Step 1. Given information

Part (a) Step 2. Find whether the given function g is even, odd, or neither.

Part (b) Step 1. Determine the point of local maximum value.

Part (b) Step 2. Determine the local maximum value at the point x=-3.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Pure Maths

Probability and Statistics

Calculus

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Logic and Functions

Study anywhere. Anytime. Across all devices.

$g (x) = x^{3} - 27 x$
(a) Determine whether the given function $g$ is even, odd, or neither.
(b) There is a local minimum value of $- 54$ at $3$ . Determine the local maximum value.