The table below gives selected values of a function f. The function is twice differentiable with f''(x)>0. Which of the following could be the value of f'(3)?

Question

This is the the table to solve the equation above

X F(x)

1. 2.4

3. 3.6

5. 5.4

Here are the answer choices

1.2

0.6

0.9

0.7

1.5

Answer 1

This question needs you to understand a bit about what derivatives tell you about a function. You've probably heard that derivatives tell you about the rate of change of a function. But since the derivative f' is also a function it makes sense to ask about its rate of change too. This, of course, is f'' (the derivative of the derivative of f.)

To say f''(x) > 0 means the slopes of the tangent lines to the graph of f are increasing as x increases (or the rate of change is getting more positive as x gets more positive.) Okay.. so how do we use that here? Well the table you have gives us a little bit about how the function is changing even if we can't compute derivatives of f explicitly from the table.

Here's what I mean: from x = 1 to x = 3 (an increase of 2 in the positive x-direction) the function f increases from 2.4 to 3.6 (the function increases by 3.6 - 2.4 = 1.2). Since the function is (twice) differentiable we can conclude that the average rate of change for f over the interval from x = 1 to x = 3 is (change in f)/(change in x) = 1.2/2 = 0.6.

Hopefully it makes sense that whatever the rate of change of f is at x = 3 it has to be greater than the rate of change between x = 1 and 3 since that rate of change was always increasing! This tells us that f'(3) > 0.6 which lets us rule out some of the choices right away.

To narrow it all the way down you just repeat this calculation with the average rate of change for f on the interval x = 3 to x = 5. The average rate of change of f on that interval should now be larger than f'(3). What do you get? There should now only be one remaining possibility.