Let a be a positive constant and define a function f by fx

Let a be a positive constant and define a function f by f(x) = a^2x^2 - x*. Find the maximum and minimum values of f on the interval (0, 2a]. Does the function f defined by f(x) = x^3 - 12x^2 have a maximum or a minimum

Solution

   f(x) = a²x² - x^{4 ,} interval [ 0 , 2a ]

this is a polynomial and so in continuous everywhere and in particular is then continuous on the given interval.   Now, we need to get the derivative so that we can find the critical points of the function.

f\'(x) = 2a²x - 4x³

It looks like we’ll have two critical points, x = 0 and x = +- a/2

Note that we actually want something more than just the critical points. We only want the critical points of the function that lie in the interval in question. Both of these do fall in the interval as so we will use both of them. That may seem like a silly thing to mention at this point, but it is often forgotten, usually when it becomes important, and so we will mention it at every opportunity to make sure it’s not forgotten.

Now we evaluate the function at the critical points and the end points of the interval

f(0) = 0 f(a/2) = a⁴/4

f(2a) = -12a⁴   f(-a/2) = 0

Absolute extrema are the largest and smallest the function will ever be and these four points represent the only places in the interval where the absolute extrema can occur.

So, from this list we see that the absolute maximum of f(+-a/2) is a⁴/4 and it occurs at (a critical point) and the absolute minimum of f(2a) is -12a⁴ which occurs at (an endpoint)