   Chapter 4.3, Problem 59E

Chapter
Section
Textbook Problem

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

To determine

To find:

The interval where the curve is concave downward

Explanation

1) Concept:

i) i) Fundamental Theorem of calculus. If f(x) is continuous on [a, b] and

gx=axf(t) dt

then g'x=f(x)

ii) Concavity test:

If f''x>0 on an interval, then f is concave upward on that interval.

If f''x<0 on an interval, then f is concave downward on that interval.

iii) Quotient rule:

If f and g are differentiable, then

ddxf(x)g(x)=gxddxfx-fxddxgxgx2

2) Given:

y=0xt2t2+t+2 dt

3) Calculation:

The givencurve is

yx=0xt2t2+t+2 dt

Differentiating both sides,

y 'x=ddx0xt2t2+t+2 dt

Let ft=t2t2+t+2

Then by fundamental theorem,

y'x=fx=x2x2+x+2

Therefore,

y'x=x2x2+x+2

Differentiate y'x,

By using quotient rule,

y''x=x2+x+2·ddxx2-x

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