Question

12 mark for review the graph of f, the derivative of the function f, is shown above. if f(0) = 0, which of the following must be true? i. f(0) > f(1) ii. f(2) > f(1) iii. f(1) > f(3)

Explanation:

Step1: Analyze f'(x) on (0,1)

On interval $(0,1)$, $f'(x) < 0$, so $f(x)$ is decreasing here.
Since $0 < 1$, $f(0) > f(1)$. Statement I is true.

Step2: Analyze f'(x) on (1,2)

On interval $(1,2)$, $f'(x) > 0$, so $f(x)$ is increasing here.
Since $1 < 2$, $f(2) > f(1)$. Statement II is true.

Step3: Analyze f'(x) on (1,3)

On interval $(1,3)$, the net area between $f'(x)$ and x-axis: the positive area (from 1 to 2) is larger than the negative area (from 2 to 3). By the Fundamental Theorem of Calculus, $f(3)-f(1)=\int_{1}^{3}f'(x)dx > 0$, so $f(3) > f(1)$. Statement III is false.

Answer:

I. $f(0) > f(1)$ and II. $f(2) > f(1)$ must be true.