   Chapter 2.6, Problem 34E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem

# Using Derivatives In Exercises 31–34, find the second derivative of the function and solve the equation f′′(x) = 0. f ( x ) = x x 2 + 3

To determine

To calculate: The second derivative f(x) and the solution of the equation f(x)=0 for the function f(x)=xx2+3.

Explanation

Given Information:

The function f(x)=xx2+3.

Formula used:

Quotient rule:

For any two differentiable functions u and v of x,

ddx[uv]=vdudxudvdxv2

Power rule:

If n is a real number, then

ddx(xn)=nxn1

Calculation:

Consider the provided function is,

f(x)=xx2+3

Use the provided function and the quotient rule ddx[uv]=vdudxudvdxv2 with u=x,v=x2+3 to calculate the first derivative as,

f(x)=ddx[xx2+3]=(x2+3)1x(2x)(x2+3)2=x2+32x2(x2+3)2=x2+3(x2+3)2

Now, use the obtained first derivative to calculate the second derivative as,

f(x)=ddx[f(x)]=ddx[x2+3(x2+3)2]

Use the quotient rule ddx[uv]=vdudxudvdxv2 and power rule ddx(xn)=nxn1 and differentiate as,

f(x)=(x2+3)2(2x)((x2

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