(a) Find the values of x for which the slope of the curvey = = In{x®)is cqual to zero.(b) Find the derivative of the functionf (x) = 2(3")at = 1,

Answered: (a) Find the values of z for which the…

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter2: Functions

Section2.4: Average Rate Of Change Of A Function

Problem 4.2E: bThe average rate of change of the linear function f(x)=3x+5 between any two points is ________.

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Question

(a) Find the values of x for which the slope of the curve
y = = In{x®)
is cqual to zero.
(b) Find the derivative of the function
f (x) = 2(3")
at = 1,

Expert Solution

Step 1

a).

Given,

$y = x \ln (x^{2})$

Slope of the curve is the derivative of the given curve.

So differentiating given equation with respect to x, we get

$\begin{array}{rcl} \frac{d y}{d x} & = & \frac{d (x)}{d x} \cdot \ln (x^{2}) + \frac{d (\ln (x^{2}))}{d x} \cdot x [∵ \frac{d (u \cdot v)}{d x} = \frac{d u}{d x} \cdot v + \frac{d v}{d x} \cdot u] \\ = & 1 \cdot \ln (x^{2}) + \frac{1}{x^{2}} \cdot \frac{d (x^{2})}{d x} \cdot x \\ = & \ln (x^{2}) + \frac{1}{x^{2}} \cdot 2 x \cdot x \\ = & \ln (x^{2}) + 2 \end{array}$

Now,

$\frac{d y}{d x} = 0 \Rightarrow \ln (x^{2}) + 2 = 0 \Rightarrow \ln (x^{2}) = - 2 \Rightarrow x^{2} = e^{- 2} [∵ \ln x = y \Rightarrow x = e^{y}] \Rightarrow x^{2} = \frac{1}{e^{2}} \Rightarrow x = \pm \frac{1}{e}$

Therefore slope of the given curve is zero when $x = - \frac{1}{e} & x = \frac{1}{e}$ .

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(a) Find the values of z for which the slope of the curve y = = In(2*) is equal to zero. (b) Find the derivative of the function f (x) = 2(3") %3D at = 1,