Let f (x) = x - e² *. Use f'(x) to find the following:(a) all critical points(b) the intervals on which f (x) is increasing and decreasing(c) the point at which the local minimum occurs(d) the point at which the local maximum occurs

Given that the function: fx=x·e2x Differentiate the given function with respect to “x” To apply the…

Answered: Let f (x) = x - e² *. Use f'(x) to find…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Let f (x) = x - e² *. Use f'(x) to find the following:
(a) all critical points
(b) the intervals on which f (x) is increasing and decreasing
(c) the point at which the local minimum occurs
(d) the point at which the local maximum occurs

Expert Solution

Step 1

Given that the function:

$f (x) = x \cdot e^{2 x}$

Differentiate the given function with respect to “x”

To apply the product rules of derivatives,

$\begin{matrix} f' (x) = \frac{d}{d x} (x \cdot e^{2 x}) \\ = x \frac{d}{d x} e^{2 x} + e^{2 x} \frac{d}{d x} x \\ = x \cdot e^{2 x} \frac{d}{d x} 2 x + e^{2 x} \cdot 1 \\ = x \cdot e^{2 x} \cdot (2) + e^{2 x} \\ f' (x) = e^{2 x} (2 x + 1) . . . . . . . . . (1) \end{matrix}$

Step 2

(a)

To calculate the critical points,

For critical points,

$\begin{matrix} f' (x) = 0 \\ e^{2 x} (2 x + 1) = 0 \\ 2 x + 1 = 0 \\ 2 x = - 1 \\ x = - \frac{1}{2} \\ e^{2 x} \neq 0 \end{matrix}$

Hence the critical points of the given function is -1/2

Step 3

To calculate the intervals on which the function is increasing or decreasing,

If the function is increasing then,

$\begin{matrix} f' (x) > 0 \\ e^{2 x} (2 x + 1) > 0 \\ 2 x + 1 > 0 \\ 2 x > - 1 \\ x > - \frac{1}{2} \\ x \in (- \frac{1}{2}, \infty) \end{matrix}$

Hence the function is increasing on the intervals $x \in (- \frac{1}{2}, \infty)$

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