JEE Advance - Mathematics (1981 - No. 25)
For all $$x$$ in $$\left[ {0,1} \right]$$, let the second derivative $$f''(x)$$ of a function $$f(x)$$ exist and satisfy $$\left| {f''\left( x \right)} \right| < 1.$$ If $$f(0)=f(1)$$, then show that $$\left| {f\left( x \right)} \right| < 1$$ for all $$x$$ in $$\left[ {0,1} \right]$$.
The problem statement is false, and there exists a counterexample function f(x).
The problem statement is true, and the result follows from Taylor's theorem with remainder term.
The problem statement is true, and the result follows from the Mean Value Theorem.
The problem statement is true, and the result follows from the Intermediate Value Theorem.
The problem statement is true, and the result follows from Rolle's Theorem.