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I have translated this question into English so excuse me if it is not proper.

I am struggling to figure out how to solve the following question any directions would be greatly appreciated

Let $F$ be the function defined as: $$F(x)=\sqrt{x-x^2}.$$

Additionally, consider $p_2(x)$, a polynomial interpolation that passes through the points $x_1$, $x_2 = 1$, and $x_3 = 0$.

Objective:

The goal is to determine the largest value of $x_1$ within the interval $(0, 1)$ that satisfies the equation: $$F(0.5) - p_2(0.5) = -0.25$$

I tried doing Lagrange interpolation, however, that leaves me with an equation that requires a root-finding method (I tried bisection) and I have not managed to find $x_1$ that satisfies the above requirement

Lutz Lehmann
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NeedsToKnow
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    To @NeedsToKnow: I edited your Q, but I am confused with your function $F(x)$. Please, check my edit... – Anton Vrdoljak Sep 16 '23 at 16:39
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    @AntonVrdoljak : This is Matlab notation, appropriate if the answer is expected in Matlab code, but not so much here for a theoretical question without mentioning Matlab. – Lutz Lehmann Sep 16 '23 at 17:06
  • To @LutzLehmann: If you know how to improve formatting regarding function $F(x)$, please do it. Thanx! – Anton Vrdoljak Sep 16 '23 at 17:10

1 Answers1

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Any quadratic polynomial with $p_2(0)=p_2(1)=0$ has the form $p_2(x)=cx(1-x)$ for some $c$.

Then for the required value at the midpoint one gets $p_2(0.5)=F(0.5)+0.25=0.75=c\cdot 0.25$, so $c=3$. With $p_2(x_1)=F(x_1)$ one gets $3\sqrt{x_1(1-x_1)}=1$, which is a quadratic equation that should be easy to solve.

Lutz Lehmann
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