Question

A function is defined as \[ f(x)= \begin{cases} 3x-1, & 0\le x\le 2,\\ \sqrt{25(x-1)}, & 2\le x<\infty. \end{cases} \] For \(f(x)\) in the interval \(\left[\frac13,3\right]\), choose the correct statement.

• A. Only Rolle's theorem is applicable, but Lagrange's Mean Value Theorem is not applicable
• B. Rolle's theorem is not applicable but Lagrange's Mean Value Theorem is applicable
• C. Both Rolle's and Lagrange's theorems are applicable
• D. Both Rolle's and Lagrange's theorems are not applicable

Hint:

For piecewise functions, always check continuity and differentiability at the junction point before applying Rolle's theorem or LMVT.

Answer 1

The Correct Option is B