How do I find the maximum of this function algebraically?

Question

How do I find the maximum of f(x)=(4x^3)-(99.06x^2)+603.2246x
Please tell me how to solve this algebraically(not with differential calculus nor graphically)
I would greatly appreciate it if you showed me the process step by step. Thank u so much =)

Answer 1

Just like your other question, you find the *minimum* (this function has no maximum) by completing the square.

If you want an actual algebraic formula:

f(x) = ax^2 + bx + c
f(x) = a[x^2 + (b/a)x] + c
f(x) = a[x^2 + (b/a)x + b^2 / 4a^2] + c - b^2/4a

f(x) = a[x - b/(2a)]^2 + c - b^2/4a

Now, let's make a single fraction out of the last two terms.

f(x) = a[x - b/(2a)]^2 + [4ac - b^2]/4a

If you ever want the minimum or maximum, just calculate

[4ac - b^2]/4a

If a is positive, it's going to be a minimum.
If a is negative, it's going to be a maximum.

Edit: I just realized you're talking about a cubic; with that said, ignore everything I said above.

Answer 2

4x^3 - 99.06x^2 + 603.224 =

x^2(4x - 99.06) + 603.224

Because 4x^3 defines that when x goes to minimum infinity, the value of function goes to minimum infinity.
When x goes to maximum infinity, the value goes to maximum infinity.

So the function has two local minimum/maximum area, one is located at x^2 = 0 --> x = 0 and the other one in 4x - 99.06 = 0 -->
x = 24.765

So one local maximum or minimum at x = 0 or x = 24.765 --> f = 603.224

Answer 3

I do not see the expression entirely.
but
it may have only a local maximum