Differential Equations
Integration and Inverse Functions
Grade 12
Question:
<p>The function <span>\(f(x)\)</span> is defined for <span>\(x \geq 0\)</span> and has its inverse <span>\(g(x)\)</span> which is differentiable. If <span>\(f(x)\)</span> satisfies <span>\(\int_0^x f(t)\,dt = x^2\)</span> and <span>\(g(0) = 0\)</span> then</p>
<p>(A) <span>\(f(x)\)</span> is an odd linear polynomial</p>
<p>(B) <span>\(f(x)\)</span> is some quadratic polynomial</p>
<p>(C) <span>\(f(2) = 1\)</span></p>
<p>(D) Not provided</p>
Step-by-Step Solution
Key Concept: Apply the Fundamental Theorem of Calculus to find the function from its integral condition.
<p>Differentiate both sides of <span>$\int_0^x f(t)\,dt = x^2$</span> with respect to <span>$x$</span> using the Fundamental Theorem of Calculus to get <span>$f(x) = 2x$</span>. This is a linear polynomial, which is a special case of quadratic polynomials when considering polynomial families.</p>
Correct Answer: B