JEE Mains · Maths · STD 12 - 6. Application of derivatives
Let \(f:\left[ {0,2} \right] \to R\) be a twice differentiable function such that \(f''\left( x \right) > 0\), for all \(x \in \left( {0,2} \right)\). If \(\phi \left( x \right) = f\left( x \right) + f\left( {2 - x} \right)\), then \(\phi \) is
- A increasing on \((0, 2)\)
- B decreasing on \((0, 2)\)
- C decreasing on \((0, 1)\) and increasing on \((1, 2)\)
- D increasing on \((0, 1)\) and decreasing on \((1, 2)\)
Answer & Solution
Correct Answer
(C) decreasing on \((0, 1)\) and increasing on \((1, 2)\)
Step-by-step Solution
Detailed explanation
\(\phi(x)=f(x)+f(2-x)\) \(\phi^{\prime}(x)=f(x)-f^{\prime}(2-x)\) ........\((1)\) since \(f^{\prime \prime}(x)>0\) \(\Rightarrow f(x)\) is increasing \(\forall x \in(0,2)\) Case \(-1:\) When \(x>2-x \Rightarrow x>1\) \(\Rightarrow \phi^{\prime}(x)>0 \forall x \in(1,2)\)…
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