Differential Equations
Differential inequality
Grade Class 12
Question:
<p>\\(f''(x)>0\\) on \\((0,2)\\). \\(\\phi(x)=f(x)+f(2-x)\\). Then \\(\\phi\\) is:</p>
<span>\(increasing on (0,1) and decreasing on (1,2)\)</span>
<span>\(decreasing on (0,1) and increasing on (1,2)\)</span>
<span>\(increasing on (0,2)\)</span>
<span>\(decreasing on (0,2)\)</span>
Step-by-Step Solution
Key Concept: Compute \phi'(x) = f'(x) - f'(2-x). Use f'' > 0.
<div class='solution'><p>\(\phi'(x)=f'(x)+f'(2-x)\cdot(-1)=f'(x)-f'(2-x)\).</p><p>Since \(f''>0\): \(f'\) is strictly increasing. For \(x<1\): \(x<2-x\) → \(f'(x)<f'(2-x)\) → \(\phi'(x)<0\): decreasing on \((0,1)\). For \(x>1\): \(x>2-x\) → \(f'(x)>f'(2-x)\) → \(\phi'(x)>0\): increasing on \((1,2)\). <strong>Answer: (2)</strong>.</p><p class='key-concept'>🔑 Key Concept: \(f''>0\) means \(f'\) is increasing (convex function). Use this to determine sign of \(\phi'\).</p></div>
Correct Answer: 2