Definite Integration
Grade None
Question:
<p>A differentiable function f is given by f(x) = <span class="math-tex">\(\frac{1}{x^{2}} \int \limits_{4}^{x}\)</span>(4t<sup>2</sup> - 2f'(t))dt, x > 0. The value of f'(4) is</p>
<p style="display:inline">16</p>
<p style="display:inline">0</p>
<p style="display:inline"><span class="math-tex">\(\frac{32}{9}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{64}{9}\)</span></p>
Step-by-Step Solution
Key Concept: Apply the product rule and the Leibniz rule to differentiate the function, then evaluate at the limit point where the integral term vanishes.
<p>f'(x) = <span class="math-tex">$\frac{1}{x^{2}}$</span>[4x<sup>2</sup> - 2f'(x)] - <span class="math-tex">$\frac{2}{x^{3}}$</span><span class="math-tex">$\int \limits_{a}^{x}$</span>(4t<sup>2</sup> - 2f'(t))dt ...[using Leibnitz's rule]<br />
<span class="math-tex">$\Rightarrow$</span> f'(4) = <span class="math-tex">$\frac{1}{16}$</span>[64 - 2f'(4)] - 0<br />
<span class="math-tex">$\Rightarrow$</span> <span class="math-tex">$\frac{9}{8}$</span>f'(4) = 4 <span class="math-tex">$\Rightarrow$</span> f'(4)] = <span class="math-tex">$\frac{32}{9}$</span></p>
Correct Answer: C