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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Finding Relative Extrema In Exercises 65-68, find all relative extrema of the function. Use a graphing utility to verify your result. See Example 6.

y = ln ( 1 4 x x 2 )

To determine

To calculate: The relative extrema of the function y=ln(14xx2).

Explanation

Given Information:

The provided function is y=ln(14xx2).

Formula used:

Rules for derivatives:

Sum rule of derivative:

(f(x)+g(x))'=f'(x)+g'(x)

Derivative of function f(x)=ln x is,

ddxlnx=1x

Constant multiple rule is f'(cx)=cf(x) where c is constant.

Derivative of function y=xn using power rule is dydx=nxn1.

If f(x) and g(x) are two functions then quotient rule of derivative of functions is,

ddx[f(x)g(x)]=g(x)f(x)f(x)g(x)[g(x)]2

Calculation:

Consider the function y=ln(14xx2).

First, evaluate the first derivative of the function y=ln(14xx2).

dydx=ddxln(14xx2)

Apply the chain rule,

ddxln(14xx2)=ddxln(14xx2)×ddx(14xx2)=1(14xx2)[ddx1ddx4xddx(x2)]=1(14xx2)[042x]=42x(14xx2)

Hence, the first derivative of the function is dydx=42x(14xx2).

Now, set the derivative equal to 0.

42x(14xx2)=042x=02x=4x=2

Hence, the only critical number is x=2

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