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Let K(t) be a measure of the knowledge you gain by studying for a test for t hours. Which do you think is larger, Is the graph of K concave upward or concave downward? Why?

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Nestor Rutherford
Nestor RutherfordLv2
23 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?

2. (20 pts) An object begins moving along a vertical line at time t = Os and stops at time t = 6 s. Its height above the ground at time t is given by h(t) in the provided graph, where h is measured in meters. Use the graph to answer the following questions. on 0 (a) Compute the average velocity of the object on the time interval [0,6). Show your work. N 1 2 3 4 5 6 (b) Circle the choice that best describes the shape of the graph on the time interval (5,6). i. increasing and concave up ii. increasing and concave down iii. decreasing and concave up iv. decreasing and concave down (c) At what time is the object farthest away from the ground? What is the velocity at that time? time= velocity= (d) The maximum velocity attained on (0,6) occurs at one of the following times. Circle the correct choice. i. ls ii. 2 s iii. 3s iv. 4s v. 5s (e) Estimate the value of the maximum velocity attained. The value is an integer. (f) Let v(t) be the velocity at time t. Circle the greatest value amongst the following choices. i. v(2.5) ii. (2.6) ii. v(2.7) iv. v(2.8) V. (2.9) (8) List the time interval(s) on which the velocity is increasing. Use integer endpoints. (h) The maximum speed attained on (0,6) occurs at one of the following times. Circle the correct choice. i. ls ii. 2s iii. 35 iv. 4s v. 5s

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