Find the value of a and the coordinates Q and T

[Saracen Rue]

Homework Statement

The function $f(x)=2a^2x\left(x-a\right)^2$ intersects with the line $y=ax$ at the origin, point $Q(b,f(b))$ and point $T(c,f(c))$ where $c>b>0$. A probability density function, $p(x)=ax-f(x)$ can be formed over the domain $[b,c]$.

(a) Determine the exact values for:

1. The constant, $a$
2. The coordinates $Q$ and $T$

(b) Calculate $Pr(2^{1/3}≤X≤2^{2/3})$ correct to 4 decimal places

Homework Equations

For PDFs: $∫_b^cf(x)dx=1$

The Attempt at a Solution

I know that for $p(x)$ to be a probability function, the integration of $p(x)$ over the domain $[b,c]$ must equal 1. However, the problem here is that we don't have the values of $b$ or $c$. Given that part 1 of the question is to find $a$, I assume we have to find that to be able to find $b$ and $c$. However, I'm kind of at a loss as how I would go about doing this.

 

[haruspex]

we don't have the values of b or c

You know they are such that Q and T lie on the intersection of the two curves, so b and c must be solutions to a certain equation.

 

[Bill_Nye_Fan]

I'll give you a small hint to get you started: the first step is to solve a couple things simultaneously. You shouldn't have too much trouble with the rest from there :)