Let f:[1, ∞) →[2, ∞) be a differentiable function. If 10 ∫₁ˣ f(t) dt=5 x f(x)-x⁵-9 for all x ≥ 1, then the value of f(3) is :

Solution

Maths sir · 4 steps

Differentiate both sides to get an equation in f, then use x = 1 to fix the constant.

1. Differentiate. By the Fundamental Theorem, the derivative of ∫₁ˣ f(t) dt is f(x). Differentiating both sides gives 10f(x) = 5f(x) + 5xf'(x) - 5x⁴, so xf'(x) - f(x) = x⁴.
2. Spot the quotient. Divide by x². The left side becomes the derivative of f(x)/x, so the equation is ready to integrate.

   d/dx(f(x)/x) = x² ⇒ f(x) = x⁴/3 + Cx

3. Find f(1). Put x = 1 in the given equation. The integral from 1 to 1 is 0, so 0 = 5f(1) - 1 - 9, giving f(1) = 2. Then 1/3 + C = 2, so C = 5/3.
4. So. f(x) = x⁴/3 + 5x/3, and f(3) = 81/3 + 15/3 = 27 + 5 = 32.

The idea to keep. When an integral equation has a variable upper limit, differentiate to get a differential equation. Then put x equal to the lower limit, where the integral becomes zero, to find the constant.

Answer from the NTA final answer key. Solution by PrepWiser.