Limits, Continuity & Differentiability
Increasing and Decreasing Functions
Grade 12
Question:
<p>Let <em>F(x)</em> be an increasing function and <em>G(x)</em> be a decreasing function. Which of the following are correct?</p><p>(a) <em>F(F(x) − G(x))</em> is an increasing function</p><p>(b) <em>G(F(x) − G(x))</em> is a decreasing function</p><p>(c) <em>G'(x) − F'(x) < 0</em></p><p>(d) <em>G'(F(x)) · F'(x) < 0</em></p>
<p>(a) <em>F(F(x) − G(x))</em> is increasing</p>
<p>(b) <em>G(F(x) − G(x))</em> is decreasing</p>
<p>(c) <em>G'(x) − F'(x) < 0</em></p>
<p>(d) <em>G'(F(x)) · F'(x) < 0</em></p>
Step-by-Step Solution
Key Concept: For composite functions, the monotonicity depends on the composition of monotonic functions: increasing∘increasing = increasing, decreasing∘increasing = decreasing. The sign of derivatives follows: F' > 0, G' < 0 (since F increasing, G decreasing).
<p><strong>Step 1: Analyze the given conditions</strong></p><p>F(x) is increasing ⟹ F'(x) > 0 everywhere</p><p>G(x) is decreasing ⟹ G'(x) < 0 everywhere</p><p><strong>Step 2: Check option (a) - F(F(x) − G(x))</strong></p><p>Let h(x) = F(x) - G(x). Then h'(x) = F'(x) - G'(x) = (positive) - (negative) > 0</p><p>So h(x) is increasing. Since F is increasing and h(x) is increasing:</p><p>F(h(x)) = F(F(x) - G(x)) is increasing. ✓ <strong>CORRECT</strong></p><p><strong>Step 3: Check option (b) - G(F(x) − G(x))</strong></p><p>Using h(x) = F(x) - G(x) where h'(x) > 0 (h is increasing)</p><p>G(h(x)) composition: G is decreasing, h(x) is increasing</p><p>By composition rule: decreasing ∘ increasing = decreasing</p><p>So G(F(x) - G(x)) is decreasing. ✓ <strong>CORRECT</strong></p><p><strong>Step 4: Check option (c) - G'(x) − F'(x) < 0</strong></p><p>G'(x) is negative and F'(x) is positive</p><p>Therefore: G'(x) - F'(x) = (negative) - (positive) < 0 ✓ <strong>CORRECT</strong></p><p><strong>Step 5: Check option (d) - G'(F(x)) · F'(x) < 0</strong></p><p>G'(F(x)) < 0 (since G is decreasing, its derivative is always negative)</p><p>F'(x) > 0 (since F is increasing)</p><p>Product: (negative) × (positive) = negative < 0 ✓ <strong>CORRECT</strong></p><p><strong>Step 6: Check option (a) again</strong></p><p>Option (a) states F(F(x) - G(x)) is increasing, which we proved TRUE.</p><p>∴ Answer: <strong>BCD</strong></p>
Correct Answer: BCD