Question

The graph of y = f '(x), the derivative of f(x), is shown below. Given f(-4) = 2, evaluate f(4).

The options are:
0
2
8
10

Answer 1

2

Explanation:

f'(x) is an odd function (symmetrical about the origin), therefore f(x) is an even function (symmetrical about the y-axis). So f(x) = f(-x).

Since f(-4) = 2, f(4) = 2.

We can also show this using integrals:

f(-4) = ∫₀⁻⁴ f'(x) dx + C

2 = ½ (-4)(-2) + C

2 = 4 + C

C = -2

f(4) = ∫₀⁻⁴ f'(x) dx − 2

f(4) = ½ (4)(2) − 2

f(4) = 4 − 2

f(4) = 2