If $\, f \in C \left[a,b\right]\, $ and $a< x_1< x_2 < b,\, $ prove $\, \exists \xi \in(a,b)\, $ s.t. $\, f(\xi) = \frac{f(x_1) + f(x_2)}{2}$

Question

If $\, f \in C \left[a,b\right]\, $ and $a< x_1< x_2 < b,\, $ prove $\, \exists \xi \in(a,b)\, $ s.t. $\, f(\xi) = \frac{f(x_1) + f(x_2)}{2}$

My attempt

Proof

Without loss of generality let $f(x_1) \geq f(x_2)$. Then we have

$$2f(x_1) \geq f(x_1) + f(x_2) \geq 2f(x_2)$$ $$f(x_1) \geq \frac{f(x_1) + f(x_2)}{2} := k \geq f(x_2)$$

Since we are given that $f$ is continuous and that $f(x_1)\geq k \geq f(x_2)$ we are guaranteed by the intermediate value theorem that there exists a $\xi \in (x_1, x_2) \subset (a,b)$ such that $f(\xi) = k$.

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Just checking if it's clear, or if I missed anything.

Answer 1

Your proof is fine. You missed nothing.