WBJEE · Maths · Application of Derivatives
Let \(\phi(x)=f(x)+f(2 a-x), x \in[0,2 a]\) and \(f^{\prime \prime}(x)\gt0\) for all \(x \in[0, a]\). Then \(\phi(x)\) is
- A increasing on \([0, a]\)
- B decreasing on \([0, a]\)
- C increasing on \([0,2 \mathrm{a}]\)
- D decreasing on \([0,2 \mathrm{a}]\)
Answer & Solution
Correct Answer
(B) decreasing on \([0, a]\)
Step-by-step Solution
Detailed explanation
\(\phi^{\prime}(x)=f^{\prime}(x)-f^{\prime}(2 a-x) \quad \because f^{\prime \prime}(x)\gt0\) \(\therefore \mathrm{f}^{\prime}(\mathrm{x})\) is increasing So, \(\phi(x)\) is decreasing in \((0, a)\) as \(f^{\prime}(x) \lt f^{\prime}(2 a-x)\) in \((0, a)\)
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