Question

F is a twice differentiable function that is defined for all reals. The value of f "(x) is given for several values of x in the table below.
x –8 –3 0 3 8
f " (x) –4 –2 0 4 5

If f " (x) is always increasing, which statement about f(x) must be true?
f(x) is concave downwards for all x.
f(x) passes through the origin.
f(x) has a relative minimum at x = 0. <- my answer
f(x) has a point of inflection at x = 0.

Answer 1

The correct answer is actually the last one.

The second derivative $f''(x)$ gives us information about the concavity of a function: if $f''(x)$ then the function is concave downwards in that point, whereas if $f''(x)>0$ then the function is concave upwards in that point.

This already shows why the first option is wrong - if the function was concave downwards for all x, then the second derivate would have been negative for all x, which isn't the case, because we have, for example, $f''(8)=5$

Also, the second derivative gives no information about specific points of the function. Suppose, in fact, that $f(x)$ passes through the origin, so $f(0)=0$. Now translate the function upwards, for example. we have that $f(x)+k$ doesn't pass through the origin, but the second derivative is always $f''(x)$. So, the second option is wrong as well.

Now, about the last two. The answer you chose would be correct if the exercise was about the first derivative $f'(x)$. In fact, the first derivative gives information about the increasing or decreasing behaviour of the function - positive and negative derivative, respectively. So, if the first derivative is negative before a certain point and positive after that point. It means that the function is decreasing before that point, and increasing after. So, that point is a relative minimum.

But in this exercise we're dealing with second derivative, so we don't have information about the increasing/decreasing behaviour. Instead, we know that the second derivative is negative before zero - which means that the function is concave downwards before zero - and positive after zero - which means that the function is concave upwards after zero.

A point where the function changes its concavity is called a point of inflection, which is the correct answer.