Integral Calculus-2
Integral Calculus-2
Allen Star Batch
Grade 12
Question:
If $f(x) = x + \int_0^x (y^2 + x^2)f(y) dy$, then:
$f(x) = \frac{260}{119}$
$f(-1) = \frac{-100}{119}$
$f(x)$ have positive point of local minimum
$f(x)$ have negative point of local minimum
Step-by-Step Solution
Key Concept: Functional equations with integrals can be solved by assuming a polynomial form and matching coefficients.
The functional equation shows $f(x)$ is quadratic: $f(x) = ax + bx^2$. Let $f(x) = ax + bx^2$ with $f'(x) = ay + by^2$. Then $a = 1 + \int_0^1 y^2(ay + by^2)dy = 1 + a\frac{1}{4} + b\frac{1}{5}$ gives $\frac{3a}{4} + \frac{b}{5} = 1$. This determines the relationship between coefficients $a$ and $b$ of the quadratic function.
Correct Answer: 2,4