question_answer

Let f be a function defined by - \[f(x)=\left\{ \begin{matrix}    \frac{\tan x}{x} & ,x\ne 0  \\    1 & ,x=0  \\ \end{matrix} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\ne 0\] Statement -1 : x = 0 is point of minima of f Statement-2 : f'(0) =0.     AIEEE  Solved  Paper (Held On 11 May  2011)

A)  Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for Statement-1.

B)  Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for statement-1

C)  Statement-1 is true, Statement-2 is false.

D)  Statement-1 is false, Statement-2 is true.

Correct Answer: B

Solution :

                \[f(x)=\left\{ \begin{matrix}    \frac{\tan x}{x} & x\ne 0  \\    1 & x=0  \\ \end{matrix} \right.\] In right neighborhood of '0'                                     tan x > x \[\frac{\tan x}{x}>1\] In left neighborhood of '0'                                              tan x < x \[\frac{\tan x}{x}>1\]                     as (x < 0) at x = 0,                                f (x) = 1 \[\Rightarrow \] x = 0 is point of minima so statement 1 is true. statement 2 obvious