## Some questions based on Rolle’s theorem

Problem 1:

Verify the truth of Rolle’s theorm for the following functions:

(a) $y=x^{2}-3x+2$ on the interval $[1,2]$.

(b) $y=x^{3}+5x^{2}-6x$ on the interval $[0,1]$.

(c) $y=(x-1)(x-2)(x-3)$ on the interval $[1,3]$.

(d) $y=\sin^{2}(x)$ on the interval $[0,\pi]$.

Problem 2:

The function $f(x)=4x^{3}+x^{2}-4x-1$ has roots 1 and $-1$. Find the root of the derivative $f^{'}(x)$ mentioned in Rolle’s theorem.

Problem 3:

Verify that between the roots of the function $y=\sqrt{x^{2}-5x+6}$ lies the root of its derivative.

Problem 4:

Verify the truth of Rolle’s theorem for the function $y=\cos^{2}(x)$ on the interval $[-\frac{\pi}{4},+\frac{\pi}{4}]$.

Problem 5:

The function $y=1-\sqrt{x^{4}}$ becomes zero at the end points of the interval $[-1,1]$. Make it clear that the derivative of the function does not vanish anywhere in the interval $(-1,1)$. 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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