Question

Values of f(x) are given at a, b, and c. Show that the maximum is obtained by

f(a) (b^2 - c^2) + f(b) (c^2 - a^2) + f(c) (a^2 - b^2)

x = -----------------------------------------------------------------

f(a) (b - c) + f(b) (c - a) + f(c) (a - b)

Answer 1

If the condition of the task is:

the maximum is obtained by

$x=\frac{f(a) (b^2 - c^2) + f(b) (c^2 - a^2) + f(c) (a^2 - b^2) }{f(a) (b - c) + f(b) (c - a) + f(c) (a - b)}$

then:

we don't know what is $f(x)$, and we can put any values for $f(a), f(b),f(c)$ ;

and we cannot say what value is of $f_{max}(x)$.

So, we have some abstract formula.

That's why I think that it should be some other additional condititions for $f(x)$ .

For example:

$a=1, b=2, c=3$

Since we don't know what is $f(x)$, we can put any values for $f(a), f(b), f(c)$

For example:

$f(a)=10, f(b)=20, f(c)=30$

So, we get:

$x=\frac{10 (4 - 9) + 20 (9 - 1) + 30 (1 - 4) }{10 (2 - 3) + 20 (3 - 1) + 30 (1 - 2)}\to \infin$

But, again, a function $f(x)$ is unknown for us. And we have not to say about the given value of $x$ or value of $f_{max}(x)$.

We have to know value(s) of $x$ when $f'(x)=0$ , but it is impossibile having the given condititions of the task.