Definite Integration
Grade None
Question:
<p>If <span class="math-tex">\(\int_\limits0^x\)</span> f(t) dt = x<sup>2</sup> + <span class="math-tex">\(\int_\limits x^1\)</span> t<sup>2</sup>f(t) dt, then <span class="math-tex">\(f^{\prime}(\frac 1 2)\)</span> is:</p>
<p style="display:inline"><span class="math-tex">\(\frac 45\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{18}{25}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{24}{25}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac 6{25}\)</span></p>
Step-by-Step Solution
Key Concept: The fundamental approach involves differentiating the given integral equation using the Leibniz Rule to transform it into an explicit functional form for f(x).
<p><span class="math-tex">$\int_\limits{0}^{x}$</span> f(t)dt = x<sup>2</sup> + <span class="math-tex">$\int_\limits{1}^{x}$</span>t<sup>2</sup> f(t)dt<br />
<span class="math-tex">$\Rightarrow$</span> f(x) = 2x - x<sup>2</sup>f(x)<br />
<span class="math-tex">$\Rightarrow f(x)=\frac{2 x}{1+x^{2}}$</span><br />
<span class="math-tex">$\Rightarrow f^{\prime}(x)=\frac{2\left(1-x^{2}\right)}{\left(1+x^{2}\right)^{2}}$</span><br />
Then,<br />
<span class="math-tex">$f^{\prime}(\frac 1 2)=\frac{2\left(1-\frac{1}{4}\right)}{\left(1+\frac{1}{4}\right)^{2}}=\frac{3}{2} \times \frac{16}{25}=\frac{24}{25}$</span></p>
Correct Answer: C