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What is the condition for two extreme points?

I am new to the topic of extreme point and trying to get the whole picture. Now I know the extreme theorm that mentions two condition to obtain a critical points what are $f'(x)= 0$ or $f'(x)=$ undefined.

I solved some examples of those on Khan Academy but coming across this example $mathrm{Let} , a,b,c,d, in , mathbb{R}, aneq0, f: mathbb{R} longrightarrow mathbb{R}, f(x)=a,x^3 + b ,x^2 + c, x + d.$

  • (a) What is the condition for two extreme points?
  • the answer supposed to be (a) $,b^2>3ac$.

Can I know how and which topic this fall into?

Mathematics Asked by Mai Ehab on November 14, 2021

1 Answers

One Answer

if you calculate de $f'(x) = 0$, you will see the condition for the "square root" to be positive (you fall into a quadratic equation), is $$sqrt{(b^2 - 3ac)},$$ so, to be positive, you'll need $$b^2 > 3ac.$$ After this you can exploit the signals of $a$ and $c$, if you want it.

Answered by Bruno on November 14, 2021

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