Answer to Question #156539 in Calculus for Tom Garland

Question

Let f be the function given by f(x)=x/(x-4)(x+2) on the closed interval [-7,7]. Of the following intervals, on which can the Mean Value Theorem be applied?

I. [-1,3] because f is continuous on [-1,3] and differentiable (-1,3)

II. [5,7] because f is continuous on [5,7] and differentiable (5,7)

III. [1,5] because f is continuous on [1,5] and differentiable (1,5)

A. None

B. I only

C. I and II only

D. I, II, and III

Answer 1

Here you can see the graph of the above function.

The solution of the above question, is option C i.e, I and II only.

I and II can be easily seen as the justifiable option as, the required conditions for applying MVT (Mean Value Theorem) is satisfied.

But, for III at the point $x=4$ we have a discontinuous graph, as $\lim\limits_{x\to4^+}f(x)=+\infin$ and $\lim\limits_{x\to4^-}f(x)=-\infin$.

So, the condition for applying MVT is not satisfied, and hence correct option is C.