Question

Let $f\left(x\right)$ be a function defined as below :

$f\left(x\right)=\{\begin{array}{c}\sin (x^2-3x),x\le 0\\ 6x+5x^2,x>0\end{array}$
Then at $x=0,f\left(x\right)$

• A. Has a local minimum
• B. Has a local maxima
• C. Is discontinuous
• D. None of the above.

Answer 1

The correct option is B Has a local maxima

$f\left(x\right)=\{\begin{array}{c}\sin (x^2-3x),x\le 0\\ 6x+5x^2,x>0\end{array}$

$f\left(0\right)=\sin \left(0\right)=0$

$f(0-h)=\sin (h^2+3h)$

$f(0+h)=6(0+h)+5{(0+h)}^2$

$=6h+5h^2$

As h is very small

$f(0+h)=6h+5h^2<f\left(0\right)$

and $f(0-h)=\sin (h^2+3h)<f\left(0\right)$

(Image)

$f\left(0\right)>f(0-h)$

$f\left(0\right)>f(0+h)$

Thus $x=0$ is a point of local maxima