I'm not really sure where to go with this problem and I'm hoping you can help.
The problem states:
Let $f(x) = \sqrt{x - x^2}$ and $P_2(x)$ be the interpolation polynomial on $x_0 = 0, x_1$, and $x_2 = 1$. Find the largest value of $x_1$ in $(0,1)$ for which $f(0.5) - P_2(0.5) = -0.25$
What I've done:
So it's obvious that we need the second degree lagrange polynomial here. Further, $f(0.5) = 0.5$.
So solving for the lagrange polynomials:
$L_0(x) = \frac{(x-x_1)(x-1)}{(0-x_1)(0-1)} = \frac{x^2-(1-x_1)x+x_1}{x_1}$ $L_1(x) = \frac{(x-0)(x-1)}{(x_1 - 0)(x_1 -1)} = \frac{x^2-x}{x_1^2 -x_1}$ $L_2(x) = \frac{(x-0)(x-x_1)}{(1-0)(1-x_1)} = \frac{x^2 - x_1x}{1-x_1}$
and we know
$P_2(x) = L_0(x)f(x_0) + L_1(x)f(x_1) + L_2(x)f(x_2)$
So
$P_2(x) = L_0(x)* 0 + L_1(x)f(x_1) + L_2(x)*0$
So
$P_2(x) = \frac{x^2-x}{x_1^2 -x_1}f(x_1)$
But at this point I dont know where to go. We know $P_2(0.5) = .75$ from the equation in the question. But if you try to solve
$.75 = \frac{-.25}{x_1^2 -x_1}f(x_1)$
You get nothing that makes sense.
Can anyone explain to me what I'm doing wrong here? Thank you!