Is p(r) = (2/R²)r the Only Solution to the Integral Uniqueness Problem?

[HyperbolicMan]

Let p be a continuous function such that for all r1, r2 in [0,R], ∫^r2_r1p(r)dr=(r2²-r1²)/R².

I'm trying to prove that p(r)=(2/R²)r.

Question: Must p be unique? I'm not sure how to prove/disprove this.

 

[Citan Uzuki]

Try showing that p(r) - 2r/R^2 must be zero. Hint: Prove that if it isn't zero, there is a nondegenerate interval [r_1, r_2] on which it is either strictly positive or strictly negative, and derive a contradiction.

 

[mathman]

Let p be a continuous function such that for all r1, r2 in [0,R], ∫^r2_r1p(r)dr=(r2²-r1²)/R².

I'm trying to prove that p(r)=(2/R²)r.

Question: Must p be unique? I'm not sure how to prove/disprove this.

Let r1 be constant and r2 variable. Take the derivative of the integral [(r2²-r1²)/R²]
with respect to r2 and rename r2 to be r.

The derivative is unique.

 

[HyperbolicMan]

Thanks for the help!