# GATE Questions & Answers of Mean Value Theorem

## What is the Weightage of Mean Value Theorem in GATE Exam?

Total 1 Questions have been asked from Mean Value Theorem topic of Calculus subject in previous GATE papers. Average marks 1.00.

Let the function

$f\left(\theta \right)=\left|\begin{array}{ccc}\mathrm{sin}\theta & \mathrm{cos}\theta & \mathrm{tan}\theta \\ \mathrm{sin}\left(\mathrm{\pi }}{6}\right)& \mathrm{cos}\left(\mathrm{\pi }}{6}\right)& \mathrm{tan}\left(\mathrm{\pi }}{6}\right)\\ \mathrm{sin}\left(\mathrm{\pi }}{3}\right)& \mathrm{cos}\left(\mathrm{\pi }}{3}\right)& \mathrm{tan}\left(\mathrm{\pi }}{3}\right)\end{array}\right|$

where $\theta \in \left[\begin{array}{cc}\frac{\pi }{6},& \frac{\pi }{3}\end{array}\right]$ and $f\text{'}\left(\theta \right)$ denote the derivative of f with respect to θ. Which of the following statements is/are TRUE?

(I) There exists $\theta \in \left[\begin{array}{cc}\frac{\pi }{6},& \frac{\pi }{3}\end{array}\right]$ such that $f\text{'}\left(\theta \right)$=0.

(II) There exists $\theta \in \left[\begin{array}{cc}\frac{\pi }{6},& \frac{\pi }{3}\end{array}\right]$ such that $f\text{'}\left(\theta \right)$≠0.