Question

Assertion :The function $f\left(x\right)=(x^3+3x-4)(x^2+4x-5)$ has a local extremum at $x=1$ Reason: f(x) is continuous & differentiable and $f^{\prime }\left(1\right)=0$

• 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. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

$f\left(x\right)=(x^3+3x-4)(x^2+4x-5)$
$\therefore$ $f\left(1\right)=0$ and $f{\left(1\right)}^{-}>f\left(1\right)<f{\left(1\right)}^{+}$
$\therefore$ Assertion (A) is correct
Again f(x) is a polynomial function & polynomial function is continuous & differentiable in the defined domain and
$f^{\prime }\left(x\right)=2(x+2)(x^3+3x-4)+3(x^2+1)(x^2+4x-5)$
$\therefore$ $f^{\prime }\left(1\right)=0$
Assertion & Reason both are correct but $f^{\prime }\left(c\right)=0$ does not imply that f has a local extrema at $x=c.$ So Reason (R) is not the correct explanation.