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Let , where is twice differentiable for all , for all , and is concave downward on and concave upward on .

  1. At what numbers does have an extreme value?

  2. Discuss the concavity of

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Casey Durgan
Casey DurganLv2
22 May 2020

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

If f and g are positive, increasing, concave upward functions on I, show that the product function f g 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 t is decreasing. Show, by giving three examples, that f g may be concave upward, concave downward, or linear. Why doesn’t the argument in parts (a) and (b) work in this case?

(a)

Given:

Functions f and g are positive, increasing, concave upward functions on I.

Hence,

Now,

Hence, the product function is concave upward.

(b)

When both the functions are decreasing and concave upward

Hence, the product function is concave upward.

(c)

When f is increasing and g is decreasing

can either positive or negative on I, depending on values of on I..

Hence, the product function may be concave upward, concave downward or linear.

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

If f and g are positive, increasing, concave upward functions on I, show that the product function f g 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 t is decreasing. Show, by giving three examples, that f g may be concave upward, concave downward, or linear. Why doesn’t the argument in parts (a) and (b) work in this case?

The graph of the derivative f ‘of a continuous function f is shown.

(a) On what intervals is f increasing? Decreasing?

(b) At what values of x does f have a local maximum? Local minimum?

(c) On what intervals is f concave upward? Concave downward?

(d) State the x-coordinate(s) of the point(s) of inflection

(e) Assuming that f (0) = 0, sketch a graph of f.

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