Answer
Answer $c$
Work Step by Step
Since $F$ is an antiderivative of $f$ it follows that:
$$F'(x)=f(x)$$
Verify each choice.
a-$$F(x)=\int_{2}^{x}2tdt$$
Using the $FTC$ it follows:
$$F'(x)=2x$$
$$f(x)=2x$$
so option $a$ is discarded.
b-$$F(x)=\int_{0}^{2}t^{2}dt$$
Using the $FTC$ it follows:
$$F'(x)=0$$
$$f(x)=0$$
so option $b$ is discarded.
c-$$F(x)=\int_{2}^{x}t^{2}dt$$
Using the $FTC$ it follows:
$$F'(x)=x^{2}$$
$$f(x)=x^{2}$$
also $$F(2)=\int_{2}^{2}t^{2}dt=0$$
Therefore, the correct choice is $c$.
