Limits, Continuity & Differentiability
Monotonicity of Composite Functions
Grade 12
Question:
<p>If composite function \(f_1(f_2(f_3(\ldots(f_n(x))\ldots)))\) (n times) is an increasing function and if r of \(f_i\)'s are decreasing functions while rest are increasing, then maximum value of function is</p>
<p>(a) \(\dfrac{n+1}{2}\) when n is an even number</p>
<p>(b) \(\dfrac{n}{2}\) when n is an odd number</p>
<p>(c) \(\dfrac{n+1}{4}\) when n is an odd number</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: For a composite function of n functions to be increasing, r (number of decreasing functions) must be even. The product r(n-r) is maximized when r equals half of n.
<p><strong>Solution:</strong> r must be an even integer because two decreasing functions are required to make it an increasing function.</p><p>Let $y = r(n-r)$</p><p><strong>When n is odd:</strong> $r = \dfrac{n+1}{2}$ or $r = \dfrac{n-1}{2}$ for maximum value of y</p><p>Maximum $(y) = \dfrac{n^2-1}{4}$, when n is odd</p><p><strong>When n is even:</strong> $r = \dfrac{n}{2}$ for maximum value of y</p><p>Maximum $(y) = \dfrac{n^2}{4}$, when n is even</p><p>∴ Answer is (c).</p>
Correct Answer: C