Functions
General
Grade 12
Question:
Let the function f : [0, 1] → ℝ be defined by f(x) = \(\frac{4^x}{4x + 2}\). Then the value of \(\int_0^{1/40} f(\frac{1}{40}) + f(\frac{2}{40}) + f(\frac{3}{40}) + \ldots + f(\frac{39}{40}) - f(\frac{1}{40})\) is _____
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>x-sin x is odd increasing. For x>0: f'(x)=(x-sin x)+x(1-cos x)>0 → strictly increasing. Odd, range=ℝ → bijective.</p><p><strong>Answer: (C) bijective</strong></p><div class="trap-box"><strong>Trap:</strong> The |x| factor preserves oddness since it multiplies another odd factor.</div><div class="key-concept"><strong>Key Concept:</strong> Odd × odd = even? No — |x| is even, (x-sin x) is odd, product is odd</div></div>
Correct Answer: 19.00