   Chapter 4.4, Problem 94E Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Solutions

Chapter
Section Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

Proof In Exercises 93 and 94, let f and g representdifferentiable functions such that f " ≠ 0   and   g " ≠ 0 .Prove that if f and g arc positive, increasing, and concave upward on the interval (a, b), then f g is also concave upward on (a, b).

To determine

To prove: The function fg are concave upward on the interval (a,b).

Explanation

Given:

The functions f and g are positive, increasing and concave upward on the interval (a,b).

Formula used:

For a function f that is twice differentiable on an open interval I,

If f''(x)>0 for all x in I, the function f is concave upwards on I.

For any function g(x) that is differentiable in the interval (a,b) and continuous in the interval [a,b],

If d(g(x))dx>0 for the interval (a,b), the function g(x) is said to be increasing in the interval [a,b].

Proof:

Define the following function:

h(x)=f(x)g(x)

Differentiate this twice to obtain:

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

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