# Thread: Show the cubic function has extreme value

1. ## Show the cubic function has extreme value

Show that the cubic $p(x) = x^3 + ax^2 + bx + c$ has extreme values iff $a^2 > 3b.$

2. ## cubics

Any cubic goes from -infinityto +infinity. The extreme values occur when there is a bump, i.e., the derivative is zero. When you take the derivative, you'll get a quadratic. If it has real roots, you've found where the bumps are. The quadratic formula should help you.

3. Originally Posted by qmech
Any cubic goes from -infinityto +infinity. The extreme values occur when there is a bump, i.e., the derivative is zero. When you take the derivative, you'll get a quadratic. If it has real roots, you've found where the bumps are. The quadratic formula should help you.
ok, so now I have $p'(x) = 3x^2 + 2ax +b$

so do I set p'(x) = 0 and find the root? But how do I solve the equation?

4. Use the quadratic formula. You know, that stuff with $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

5. Originally Posted by 450081592
ok, so now I have $p'(x) = 3x^2 + 2ax +b$

so do I set p'(x) = 0 and find the root? But how do I solve the equation?
$p'(x)$ needs to have two real roots ... the discriminant, $b^2-4ac > 0$

$(2a)^2 - 4(3)(b) > 0$

$4a^2 - 12b > 0$

$a^2 - 3b > 0$

$a^2 > 3b$