Question

Assertion :Consider the function $f\left(x\right)=\{\begin{array}{c}-\frac{x}{2},x<0\\ 7x+8x\ge 0\end{array};f\left(x\right)$ has local minima at $x=0$. Because Reason: If $f\left(a\right)<f(a-h)$ and $f\left(a\right)<f(a+h)$ where ${}^{\prime }{h}^{\prime }$ is sufficiently small, then $f\left(x\right)$ has local minima at $x=a$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is D Assertion is incorrect but Reason is correct

Clearly, $f\left(a\right)<f(a-h)$ and $f\left(a\right)<f(a+h)$

Thus f(x) has a local minima at $x=a$ if above holds

And

$f\left(x\right)=\{\begin{array}{c}-\frac{x}{2},x<0\\ 7x+8,x\ge 0\end{array}$

$f\left(0\right)<f(0-h)$ and$f\left(0\right)<f(0+h)$

$f\left(0\right)=7\times 0+8=8$

$f(0-h)=h/8$

Since h is sufficiently small

$8\nless \frac{h}{8}$

Therefore Assertion is incorrect but reason is correct.