Problem

Match each function with its derivative.

F1

F2

F3

D1

D2

D3

Solution

F1 We notice that in this function, there is only one local extreme point, and the function is decreasing for x < 0 and increasing for x > 0. Therefore its derivative is 0 at x = 0, negative for x < 0, and positive for x > 0. This can only be the graph of D2.

F2 For this function, two characteristics stand out: this graph has the most extrema (and therefore its derivative must have the most roots), and the slopes of the function change sign more quickly near the origin (so the derivative should cross the x axis more often close to the origin). This behavior is consistent with graph D3.

F3 Besides using elimination to see that D1 must go with this function, notice that F3 has 3 local extreme points, so its derivative will have 3 zeros.