Please watch this video to solve this question efficiently and clearly.
Direct TimeStamp of Detailed Video Solution
Since $g(x)$ is a polynomial function it is continuous and differentiable everywhere. Therefore there are two theorems that we might use to answer this question. The first is the Intermediate Value Theorem, which says that between -1 and 2 and any $y$-value between 4 and 7 there is at least one number $c$ such that $g(c)$ is equal to that $y$-value. Since none of the answer choices involve $y$-values between 4 and 7, we go on to the next theorem.
The Mean Value Theorem says that between -1 and 2 there is at least one number $c$ such that $g^{\prime}(c)$ is equal to the slope of the secant line between the points $(-1,4)$ and $(2,7)$, i.e. 1. So (b) is correct.
There is nothing else we can conclude from the information we are given.
$$g^{\prime}(c)=\frac{g(2)-g(-1)}{2-(-1)}=\frac{7-4}{3}=1$$
