Question
Let f:[1, ) R be a differentiable function. If 6 _ 1 ^ x f(t) d t=3 x f(x)+x^ 3 -4 for all x 1, then the value of f(2)-f(3) is
JEE Main
Mathematics
Differential Equations
2026
JEE Main 2026 (22 January Shift 1)
JEE Main Mathematics Question (2026) — Solution
Options
- A. -4
- B. 3
- C. 4
- D. -3
Answer
B. 3
Step-by-step solution
Differentiating 6 _1^x f(t)dt = 3xf(x) + x^3 - 4: 6f(x) = 3f(x) + 3xf'(x) + 3x^2 f(x) - xf'(x) = x^2. d dx ( f(x) x ) = xf'(x)-f(x) x^2 = -1. Integrating: f(x) x = -x + C f(x) = -x^2 + Cx. At x = 1: 0 = 3f(1) - 3 f(1) = 1. So -1 + C = 1 C = 2. f(x) = -x^2 + 2x. f(2) = 0, f(3) = -3. f(2) - f(3) = 3.
Related: Mathematics — Differential Equations · All PYQ Banks