What is the value of the parameter a?

Question

It is given a function f(x) = $2x^{3} - 9ax^{2} + 12a^{2}x + 1$, if $x_1$=max and $x_2$=min and $x_1^{2}$=$x_2$, then what is the value of the parameter a

Answer 1

What you should do is calculate the derivative of $f(x)$. With that you should calculate the zeros of $f^\prime(x)$ so you can find the maximum and minimum of $f(x)$. With that, you can find what is $x_1$ and $x_2$. All this will be done in function of $a$, whose value you do not know. When you make $x_1^2 = x_2$ you will have an equation solvable for $a$.

Having found that the roots of $f^\prime(x) $ are $a$ and $2a $, you should see which one ($2a $ or $a $) is the point where the function attains a maximum. That is $x_1$. The other will be $x_2$. Then you get an equation with only one unknown, $a $, because $x_1^2 = x_2$.

Using the original expression for $f $ we find $f(2a) = 4a^3 + 1$ and $f(a) = 5a^3 + 1$.

Which one is bigger?

If $a $ is positive then $a$ is the maximum and $2a $ is the minimum. If $a $ is negative it is the other way around.

You must use the condition $x_1^2 = x_2$ to find if $a $ is positive or negative.

Then solve $x_1^2 = x_2$ for $a $

We can either have $x_1 = a$ or $x_1 = 2a$. Either way, and because $x_1^2 > \geq 0$, we know that $a$ has to be positive for the equation $x_1^2 = x_2$ to make sense.

If $a > 0$, then $a$ is the maximum and $x_1 = a, x_2 = 2a$. Therefore we get $a^2 = 2a$. Can you solve this now?