Question

Assertion :$f\left(x\right)$ is increasing with concavity upwards, then concavity of $f^{-1}\left(x\right)$ is also upwards. Reason: If $f\left(x\right)$ is decreasing function with concavity upwards, then concavity of $f^{-1}\left(x\right)$ is also upwards.

• 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 and Reason are correct

Answer 1

The correct option is D Assertion is incorrect and Reason are correct

Let $g\left(x\right)$ be the inverse function of $f\left(x\right).$

Then $f\left(g\left(x\right)\right)=x$

$\therefore f^{\prime }\left(g\left(x\right)\right).g^{\prime }\left(x\right)=1\Rightarrow g^{\prime }\left(x\right)=\frac{1}{f^{\prime }\left(g\left(x\right)\right)}$

$\Rightarrow g^{\prime \prime }\left(x\right)=-\frac{1}{f^{\prime \prime }\left(g\left(x\right)\right)}.g^{\prime }\left(x\right)$

In assertion $f^{\prime \prime }\left(g\left(x\right)\right)>$ and $g^{\prime }\left(x\right)>0$

$\therefore g^{\prime \prime }\left(x\right)<0$ and the concavity of $f^{-1}\left(x\right)$ is downwards

$\therefore$ assertion is incorrect.

In reason $-2f^{\prime \prime }\left(g\left(x\right)\right)>0$ and $g^{\prime }\left(x\right)<0$

$\therefore g^{\prime \prime }\left(x\right)>0$ and so the concavity of $f^{-1}\left(x\right)$ is upwards

$\therefore$ reason is correct.