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 :)