Question

Let $f$ be a function defined by $f\left(x\right)=\{\begin{array}{c}\frac{\tan x}{x},x\ne 0\\ 1,x=0\end{array}$
Statement 1: $x=0$ is a point of minima of $f$
Statement 2: $f^{\prime }\left(0\right)=0$

• A. Statement 1 is false, statement 2 is true
• B. Statement 1 is true, statement 2 is true, statement 2 is correct explanation for statement 1
• C. Statement 1 is true, statement 2 is true, statement 2 is not correct explanation for statement 1
• D. Statement 1 is false, statement 2 is false

Answer 1

The correct option is B Statement 1 is true, statement 2 is true, statement 2 is correct explanation for statement 1

$f\left(x\right)=\frac{\tan x}{x}$

$f^{\prime }\left(x\right)=\frac{x-\frac{1}{2}\sin 2x}{x^2{\cos }^{2}x}$

$f^{\prime }\left(0\right)=\underset{x\to 0}{lim}\frac{x-\frac{1}{2}\sin 2x}{x^2{\cos }^{2}x}=\underset{x\to 0}{lim}\frac{1-\cos 2x}{2x}\frac{1}{{\cos }^{2}x}=0$

$f^{\prime \prime }\left(x\right)=\frac{x^2{\sin }^{2}x{\cos }^{2}x-(x-\frac{1}{2}\sin 2x)(2x{\cos }^{2}x-x^2\sin x)}{x^4{\cos }^{4}x}$

$f^{\prime \prime }\left(0\right)=1>0⟹$ at $x=0$ is a point minima of $f$