Question

Let \(\varphi(x)=f(x)+f(1-x)\) and \(f^{\prime \prime}(x)<0\) in \([0,1]\), then
(A) \(\varphi\) is monotonic increasing in \(\left[0, \frac{1}{2}\right]\) and monotonic decrasing in \(\left[\frac{1}{2}, 1\right]\)
(B) \(\varphi\) is monotonic increasing in \(\left[\frac{1}{2}, 1\right]\) and monotonic decrasing in \(\left[0, \frac{1}{2}\right]\)
(C) \(\varphi\) is neither increasing nor decreasing in any sub interval of \([0,1]\)
(D) \(\varphi\) is increasing \([0,1]\)

Answer 1

Ans: (A)
Hint : \(\phi^{\prime}(x)=f^{\prime}(x)-f^{\prime}(1-x)\)
\(f^{\prime}(x)-f^{\prime}(1-x) \geq 0\) (for monotonic increasing)
\(f^{\prime}(x) \geq f^{\prime}(1-x), x \leq 1-x\left(\because f^{\prime}(x)\right.\) is decreasing \()\)
\(x \leq \frac{1}{2} \Rightarrow \phi(x)\) is monotonic increasing in \(\left[0, \frac{1}{2}\right]\) and monotonic decreasing in \(\left[\frac{1}{2}, 1\right]\)

Answer 2

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Answer 3

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