Definite Integration
Fundamental Theorem and Function Properties
Grade 12
Question:
<p>The function <span class="latex">f(x) = \int_0^x \sqrt{1 - t^4} dt</span> is such that</p>
<p>(a) it is defined on the interval <span class="latex">[-1, 1]</span></p>
<p>(b) it is an increasing function</p>
<p>(c) it is an odd function</p>
<p>(d) the point <span class="latex">(0, 0)</span> is the point of inflection</p>
Step-by-Step Solution
Key Concept: Use the fundamental theorem of calculus to find the derivative and analyze properties like monotonicity and oddness
<p><strong>Solution:</strong></p><p><span class="latex">f'(x) = \sqrt{1 - x^4} \geq 0</span> in <span class="latex">(-1, 1)</span></p><p><span class="latex">f'(x) > 0</span> for <span class="latex">x \in (-1, 1)</span>, so <span class="latex">f</span> is increasing.</p><p><span class="latex">f(x) + f(-x) = \int_0^x \sqrt{1-t^4} dt + \int_0^{-x} \sqrt{1-t^4} dt = 0</span></p><p>This shows <span class="latex">f</span> is odd.</p><p><strong>Hence, (a), (c) and (d) are the correct answers.</strong></p>
Correct Answer: a, b, c, d