Definite Integration
Functional Equations
Grade 12
Question:
<p>If <span>\(f : \mathbb{R} \to \mathbb{R}\)</span> is continuous and differentiable function such that <span>\(\int_0^x f(t) dt + \int_0^x t^3 dt = \int_0^x t^2 dt + f(1) \int_0^x f(t) dt - f(2) \int_0^x t dt + f(3)\)</span>, then the value of <span>\(f(4)\)</span> is</p>
<p>(A) 48 - 8f(1) + f(2)</p>
<p>(B) 48 - 8f(1) - f(2)</p>
<p>(C) 48 + 8f(1) + f(2)</p>
<p>(D) none of these</p>
Step-by-Step Solution
Key Concept: Differentiate the integral equation to obtain information about the function f and its values at specific points.
<p>Differentiate both sides with respect to x to eliminate the integrals and obtain a functional equation for f(x). Evaluate at specific points to solve for f(4).</p>
Correct Answer: A