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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Assume that all of the functions are twice differentiable and the second derivatives are never 0.

(a) If f and g are positive, increasing, concave upward functions on I, show that the product function fg is concave upward on I.

(b) Show that part (a) remains true if f and g are both decreasing.

(c) Suppose f is increasing and g is decreasing. Show, by giving three examples, that fg may be concave upward, concave downward, or linear. Why doesn’t the argument in parts (a) and (b) work in this case?

(a)

To determine

To show: The product function fg is concave upward that is (fg)>0 on I if f and g are positive, increasing and concave upward on I.

Explanation

Proof:

Given:

The functions f and g are twice differentiable and the second order derivative are never zero.

The functions f and g are positive on I that is, f>0 and g>0 .

The functions f and g are increasing on I that is, f>0 and g>0 .

The functions f and g are concave upward on I that is, f>0 and g>0 .

Calculation:

Consider the product of the function, h=fg .

Obtain the derivative of h.

h=(fg)

h=fg+gf

Apply the Product Rule and find the derivative of h

(b)

To determine

To show: The product function fg is concave upward that is (fg)>0 on I if f and g are positive, decreasing and concave upward on I.

(c)

To determine

To show: The product function fg may concave upward, concave downward or linear using examples.

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