What is the condition that the cubic y=ax³+bx²+cx+d shall have two Extremes?

Answer nya po is b²-3ac>0

Salamat po.

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Differentiate the given cubic equation and equate to zero. Solve the roots of the resulting quadratic equation. To have two extremes, the roots of the quadratic equation must be real. What makes the roots real is when the discriminant of the quadratic equation greater than zero.

If you have further question, don't hesitate to ask.

D ko po makuha yung roots sir

$y = ax^3 + bx^2 + cx + d$

$y' = 3ax^2 + 2bx + c = 0$

The roots of the quadratic equation are

$x = \dfrac{-B \pm \sqrt{B^2 - 4AC}}{2A}$

$x = \dfrac{-2b \pm \sqrt{4b^2 - 12ac}}{6a}$

To have two extremes

$4b^2 - 12ac \gt 0$

Yan na po value ni x?

Paano po makuha c 4b²-12ac?

Substitute:

A = 3aB = 2b

C = c

When i substitute x to y", answer nya po is 2√b²-3ac and -2√b²-3ac?

Pero yung answer daw po is only b²-3ac.

How po?

You don't need to do that. Divide 4

b^{2}- 12ac> 0 both sides by 4 and you will get the answer.Pero paano po nakuha yung 4b²-12ac?

When i substitute A,B and C

The answer is -2b+-√4b²-12ac/6a

Paano po naging 4b²-12ac nalang po?

D ko parin po makuha kung paano naging b²-3ac nalang

The problem ask a condition for the cubic to have two extremes. You give "4

b^{2}- 12ac> 0" as your condition to satisfy the problem.From $x = \dfrac{-2b \pm \sqrt{4b^2 - 12ac}}{6a}$

xis imaginary. (no extreme)x. (one extreme)x. (two extremes)That is why you give $4b^2 - 12ac \gt 0$ as you answer to satisfy the condition of the problem.

Okay na po. Salamat po .