Let f:[0,infinity)-R be a continuous strictly increasing function such that f(x)³ = integral from 0 to x of [t * f(t)²] dt for all x=0. Which of the following are INCORRECT? (A) f(x) is an onto function. (B) Integral from 0 to 1 of f(x) dx = 1/18. (C) Number of solutions of 6f(x) = e^x i

Question

Let f:[0,infinity)->R be a continuous strictly increasing function such that f(x)³ = integral from 0 to x of [t * f(t)²] dt for all x>=0. Which of the following are INCORRECT? (A) f(x) is an onto function. (B) Integral from 0 to 1 of f(x) dx = 1/18. (C) Number of solutions of 6f(x) = e^x is 2. (D) The graph of y = 6f(x) intersects the graph of y = 3x² + 2x + 2 at exactly one point.

Accepted Answer

(A) f(x) is onto function. Differentiating f³ = integral t*f² dt: 3f² f' = xf², so f'(x) = x/3. Integrating: f(x) = x²/6 (since f(0)=0). Range is [0,∞), not all of R, so f is NOT onto R — (A) is incorrect. For (B): integral(0 to 1) x²/6 dx = 1/18 — TRUE (B is correct). For (C): 6*(x²/6) = x² = e^x has 0 solutions (x² < e^x for all real x) — so (C) says 2 solutions, which is INCORRECT. For (D): 6*(x²/6) = x² vs 3x²+2x+2: x² = 3x²+2x+2 -> 2x²+2x+2=0 -> discriminant = 4-16 <0 -> no intersection — (D) says one point, INCORRECT. So A, C, D are all incorrect.

Solution

Related JEE Advanced Maths questions