   Chapter 5.3, Problem 65E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# On what interval is the curve y = ∫ 0 x t 2 t 2 + t + 2   d t concave downward?

To determine

The interval when the curve lies in concave downward.

Explanation

Given information:

The interval function is y=0xt2t2+t+2dt

Theorem of fundamental calculus part 1 as shown in below.

F(x)=axf(t)dtF(x)=f(x)

The theorem of fundamental calculus (part 1) is valid, if f is continuous on [a,b].

Calculation:

The function of y is y=0xt2t2+t+2dt (1)

Apply part 1 of theorem of fundamentals of calculus in Equation (1).

F(x)=axf(t)dty=0xt2t2+t+2dt

Find F(x) using part 1 of theorem of fundamentals of calculus:

F(x)=f(x)y=x2x2+x+2 (2)

Differentiate both sides of Equation (2) with respect to x

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Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 