Some questions based on Rolle’s theorem
Problem 1:
Verify the truth of Rolle’s theorm for the following functions:
(a) 
on the interval ![[1,2]](https://s0.wp.com/latex.php?latex=%5B1%2C2%5D&bg=ffffff&fg=000&s=0 "[1,2]")
.
(b) 
on the interval ![[0,1]](https://s0.wp.com/latex.php?latex=%5B0%2C1%5D&bg=ffffff&fg=000&s=0 "[0,1]")
.
(c) 
on the interval ![[1,3]](https://s0.wp.com/latex.php?latex=%5B1%2C3%5D&bg=ffffff&fg=000&s=0 "[1,3]")
.
(d) 
on the interval ![[0,\pi]](https://s0.wp.com/latex.php?latex=%5B0%2C%5Cpi%5D&bg=ffffff&fg=000&s=0 "[0,\pi]")
.
Problem 2:
The function 
has roots 1 and 
. Find the root of the derivative 
mentioned in Rolle’s theorem.
Problem 3:
Verify that between the roots of the function ![y=\sqrt[3]{x^{2}-5x+6}](https://s0.wp.com/latex.php?latex=y%3D%5Csqrt%5B3%5D%7Bx%5E%7B2%7D-5x%2B6%7D&bg=ffffff&fg=000&s=0 "y=\sqrt[3]{x^{2}-5x+6}")
lies the root of its derivative.
Problem 4:
Verify the truth of Rolle’s theorem for the function 
on the interval ![[-\frac{\pi}{4},+\frac{\pi}{4}]](https://s0.wp.com/latex.php?latex=%5B-%5Cfrac%7B%5Cpi%7D%7B4%7D%2C%2B%5Cfrac%7B%5Cpi%7D%7B4%7D%5D&bg=ffffff&fg=000&s=0 "[-\frac{\pi}{4},+\frac{\pi}{4}]")
.
Problem 5:
The function ![y=1-\sqrt[5]{x^{4}}](https://s0.wp.com/latex.php?latex=y%3D1-%5Csqrt%5B5%5D%7Bx%5E%7B4%7D%7D&bg=ffffff&fg=000&s=0 "y=1-\sqrt[5]{x^{4}}")
becomes zero at the end points of the interval ![[-1,1]](https://s0.wp.com/latex.php?latex=%5B-1%2C1%5D&bg=ffffff&fg=000&s=0 "[-1,1]")
. Make it clear that the derivative of the function does not vanish anywhere in the interval 
. Explain why Rolle’s theorem is NOT applicable here.
Calculus is the fountainhead of many many ideas in mathematics and hence, technology. Expect more beautiful questions on Calculus !
-Nalin Pithwa
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