Assertion \) is increasing with concavity upwards, then concavity of \) is also upwards. Reason If \) is decreasing function with concavity upwards, then concavity of \) is also upwards

Assertion is incorrect and Reason are correct

Maths · Applications of derivatives: Rate of change of quantities, monotonic-Increasing and decreasing functions, Maxima and minima of functions of one variable

Assertion \) is increasing with concavity upwards, then concavity of \) is also

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

Step-by-step solution

The second derivative of the inverse function is given by (f^{-1})''(x) = -f''(f^{-1}(x)) / [f'(f^{-1}(x))]^3. For an increasing function, f'>0, so denominator positive, making (f^{-1})'' opposite in sign to f''. Thus if f is increasing and concave up (f''>0), then f^{-1} is concave down. Hence Assertion is false. For a decreasing function, f'<0, so denominator negative, and (f^{-1})'' has same sign as f''. Thus if f is decreasing and concave up, f^{-1} is also concave up. Hence Reason is correct. Therefore, Assertion is incorrect and Reason is correct.

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Step-by-step solution

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