Let be a function defined by `f(x)={{:(tanx/x", "x ne0),(1", "x=0):}`
Statement-1: x=0 is a point on minima of f
Statement-2: f'(0)=0

Options

A. Statement-1 is True, Statement-2 is True,Statement -2 is a correct explanation for Statement -7
B. Statement -1 True ,Statement -2 is True ,Stament -2 is not a correct explanation for Statement -!
C. Statement -1 is True Statement -2 is False
D. Statement -1 is Flase,Statement -2 is True

Answer

B

Explanation

We observe that
`f'(x)=lim_(xrarr0)(f(x)-f(0))/(x-0)=lim_(xrarr0)(tanx/x-1)/x=lim_(xrarr0)(tanx-x)/x^2`
`=lim_(xrarr0)(sec^2x-1)/(2x)=1/2lim_(xrarr0)(tan^2x)/x=0`
So, statement-2 is correct
Now, `f(x)=tanx/x rArrf'(x)=(xsec^2x-tanx)/x^2=(2x-sin2x)/(2x^3cos^2x)`
As `sinx ltx" for "xgt0 and xlt sinx" for "xgt0` . Therefore , f(x) changes its sign from negative to positive as x passes through 0. Hence , x=0 is point of local minimum of f(x)
Thus , statement-1 is also correct
Clearly , statement -2 is not a correct explanation for statement-1