Limits, Continuity & Differentiability
Functional Equations
Grade 12
Question:
<p>Let <span class="math-tex">f: \mathbb{R} \to \mathbb{R}</span> is a function satisfying <span class="math-tex">f(x + y^3) = f(x) + [f(y)]^3</span> for all <span class="math-tex">x, y \in \mathbb{R}</span>. If <span class="math-tex">f'(0) \geq 0</span>, then <span class="math-tex">f(10)</span> is</p>
<p>(a) 8</p>
<p>(b) 9</p>
<p>(c) 0 or 10</p>
<p>(d) 2 or 4</p>
Step-by-Step Solution
Key Concept: Functional equations of this form are satisfied by linear functions. Determine which solutions satisfy the derivative condition f'(0) ≥ 0.
<p><strong>Step 1:</strong> Put <span class="math-tex">x = y = 0</span>:</p><p><span class="math-tex">f(0) = f(0) + [f(0)]^3 \Rightarrow [f(0)]^3 = 0 \Rightarrow f(0) = 0</span></p><p><strong>Step 2:</strong> The functional equation <span class="math-tex">f(x + y^3) = f(x) + [f(y)]^3</span> suggests linear behavior.</p><p><strong>Step 3:</strong> Try <span class="math-tex">f(x) = cx</span> for some constant <span class="math-tex">c</span>:</p><p><span class="math-tex">c(x + y^3) = cx + (cy)^3 \Rightarrow cx + cy^3 = cx + c^3y^3 \Rightarrow c = c^3 \Rightarrow c(c^2-1) = 0</span></p><p><strong>Step 4:</strong> So <span class="math-tex">c = 0, 1, -1</span>.</p><p><strong>Step 5:</strong> From <span class="math-tex">f'(0) \geq 0</span>: For <span class="math-tex">f(x) = cx</span>, we have <span class="math-tex">f'(0) = c \geq 0</span>.</p><p><strong>Step 6:</strong> This gives <span class="math-tex">c = 0</span> or <span class="math-tex">c = 1</span>.</p><p><strong>Step 7:</strong> Therefore:</p><p>If <span class="math-tex">f(x) = 0</span>, then <span class="math-tex">f(10) = 0</span></p><p>If <span class="math-tex">f(x) = x</span>, then <span class="math-tex">f(10) = 10</span></p><p>∴ Answer is (c) 0 or 10</p>
Correct Answer: C