Question

a. determine whether the mean value theorem applies to the function ( f(x)=1 + x^{2} ) on the interval ( -1,2 ).
b. if so, find the point(s) that are guaranteed to exist by the mean value theorem.
a. choose the correct answer below.
○ a. yes, because the function is continuous on the interval ( -1,2 ) and differentiable on the interval ( (-1,2) ).
○ b. no, because the function is differentiable on the interval ( (-1,2) ), but is not continuous on the interval ( -1,2 ).
○ c. no, because the function is continuous on the interval ( -1,2 ), but is not differentiable on the interval ( (-1,2) ).
○ d. no, because the function is not continuous on the interval ( -1,2 ), and is not differentiable on the interval ( (-1,2) ).

Explanation:

Step1: Check continuity

Polynomial functions \(y = 1 + x^{2}\) are continuous everywhere. So \(f(x)=1 + x^{2}\) is continuous on the closed interval \([-1,2]\).

Step2: Check differentiability

The derivative of \(f(x)\) using the power rule \((x^{n})^\prime=nx^{n - 1}\) is \(f^\prime(x)=2x\). Since \(f^\prime(x)\) exists for all real \(x\), \(f(x)\) is differentiable on the open interval \((-1,2)\).

Answer:

A. Yes, because the function is continuous on the interval \([-1,2]\) and differentiable on the interval \((-1,2)\)