Question

If the derivative of f is given by f'(x) = e^x - 3x^2, at which of the following values of x does f have a relative maximum value?

a. -0.46
b. 0.20
c. 0.91
d. 0.95
e. 3.73

Answer 1

The easiest way to solve this problem would be to graph the derivative. When the derivative equals zero, f could possibly have a local minimum or maximum at that point. Upon graphing, you'll see that f' intersects the x-axis 3 times, at
\[x = -0.459, x = 0.910, x= 3.733\] At the first and third points, the derivative changes signs from negative to positive. That means that f changes from decreasing to increasing, which would mean that those are local minima. At the middle point, f' changes from positive to negative, indicating that f changes from increasing to decreasing at that point. That would mean that the local maximum is at x = 0.91. Choice C.