Let `f(x)` be a function defined as follows: `f(x)=sin(x^2-3x),xlt=0; a n d6x+5x^2,x >0` Then at `x=0,f(x)` (a)has a local maximum (b)has a local minimum (c)is discontinuous (d) none of these

Options

A. has a local maxima
B. has a local minimum
C. is discontinuous
D. none of these

Answer

B

Explanation

We have
`underset(xrarro^-)lim f(x)= underset(xrarro^-)limsin(x^2-3x)=sin=0`
`underset(xrarro^+)lim f(x)= underset(xrarro^+)lim6x+5x^2=0, and f(0)=0`
Clearly, `underset(xrarro^-)lim f(x)= underset(xrarro^+)lim=f(0)`
So , f(x) is continious at x=0
Now, In the left neighbourhood of x=0 , we have
`f'(x)=(2x-3)cos(x^2-3x)lt0`
`rArr f(x) ` is decreasing neighbourhood of x=0 , we have
`f'(x)=6+10x gt 0`
`rArr` f(x) is increasing in the right nbd minimum .