Verify Lagrange’s mean value theorem for the following functions: f(x) = x^2 – 3x – 1, x ∈ [-11/7, 13/7]

Question

Verify Lagrange’s mean value theorem for the following functions:

f(x) = x² – 3x – 1, x ∈ \(\left[\frac{-11}{7},\,\frac{13}{7} \right]\)[-11/7, 13/7]

Answer 1

The function f given as f(x) = x² – 3x – 1 is a polynomial function.

Hence, it is continuous on \(\left[\frac{-11}{7},\,\frac{13}{7} \right]\) and differentiable on \(\left(\frac{-11}{7},\,\frac{13}{7} \right)\).

Thus, the function f satisfies the conditions of LMVT.

∴ there exists c ∈ \(\left(\frac{-11}{7},\,\frac{13}{7} \right)\) such that

and f\(\left(\frac{13}{7}\right) = \left(\frac{13}{7}\right)^2\) \(-3\left(\frac{13}{7}\right) - 1\) \(=\frac{169}{49} = \frac{39}{7} - 1\)

Hence, Lagrange’s mean value theorem is verified.