Question

Find a function that satisfies the given conditions:
f(0) = f(1) =0;
integral f(x)dx from 0 to 1 = 2;
f(x) >=0 for all x
f'(x) exists for all x

Answer 1

start by looking at the function \[g(x)=x^2(1-x)^2\]

Answer 2

you will gave to modify \(g\) a little to get the function \(f\) that you want.

Answer 3

Zarkon, how did you arrive at that initial function, please?

Answer 4

using the first conditions (f(0) = f(1) =0) i thought of

g(x)=x(1-x)

you also wanted the function to be non-negative for all x so I squared the function

it is a pollynomial so it is differentiable.

the only part to do is to make it integrate to 2

Answer 5

so find a constant c such that

\[\int_{0}^{1}cx^2(1-x)^2dx=2\]

Answer 6

then you will have \(f\)

Answer 7

Sweet! I'll try that. What I'm really trying to arrive at is a function which gives the minimal arc length for a curve which satisfies those conditions. Thanks.