Finding local max, min, and points of inflection

Find all local maximum points, local minimum points, and points of inflection on the curve y=-3x^{4}-8x^{3}+18x^{2}

What I did: First derivative is y'=-12x^{3}-24x^{2}+36x

Then I set it equal to 0

Then I factored it out: -12x(x^{2}+2x-3) = -12x(x-1)(x-2), so the critical numbers are x=0, x=1, and x=2

After getting the critical numbers, I did the second derivative

y''=36x^{2}-58x+36

I used the second derivative test, and they end up all positive, so it's all min,

So I'm a bit confused.

Isn't this the way to find **absolute minimum**, **instead of local minimum** o.o

And how do I go on from there :o

Re: Finding local max, min, and points of inflection

Quote:

Originally Posted by

**Chaim** Find all local maximum points, local minimum points, and points of inflection on the curve y=-3x^{4}-8x^{3}+18x^{2}

What I did: First derivative is y'=-12x^{3}-24x^{2}+36x

Then I set it equal to 0

Then I factored it out: -12x(x^{2}+2x-3) = -12x(x-1)(x-2), so the critical numbers are x=0, x=1, and x=2

After getting the critical numbers, I did the second derivative

y''=36x^{2}-58x+36

I used the second derivative test, and they end up all positive, so it's all min,

So I'm a bit confused.

Isn't this the way to find **absolute minimum**, **instead of local minimum** o.o

And how do I go on from there :o

There is no way that all three critical points can possibly be minima.

To answer your other question, absolute maxima and minima can occur EITHER at critical points OR at endpoints of your function.

Re: Finding local max, min, and points of inflection

Oh oops, I factored it wrong

It's actually 12x(x+3)(x-1)

So x=0, -3, and 1

So the 2nd derative would be:

-36x^{2}-58x+36

Plugging in 0 would make it 36, (So Minimum)

Plugging in 1 would make it Negative (So Maximum)

Plugging in -3 would make it Positive (So Minimum)

Though, I'm a bit confused in the multiple choice answers, because x=-3, 1 is supposed to match up into a max or min, while x=0 is alone either a max or min.

If I use the factored one: -2(18x^{2}+24x-18)

(Or maybe the multiple choice answers could be wrong as well)

Points of inflection would be where 2nd derivative is equal to 0 right?

So I did the quadratic formula within the paranthesis, and I got something different: (-29+ or - sqrt2137)/36

While in the multiple choice answers, it was like either (-2 + or - sqrt13)/3 or (-2 + or - sqrt19)/3

Re: Finding local max, min, and points of inflection

$\displaystyle y'' = -36x^2 - 48x + 36$

$\displaystyle y'' = -12(3x^2 + 4x - 3)$

$\displaystyle x = \frac{-4 \pm \sqrt{40}}{6} = \frac{-2 \pm \sqrt{10}}{3}$

Re: Finding local max, min, and points of inflection

Quote:

Originally Posted by

**skeeter** $\displaystyle y'' = -36x^2 - 48x + 36$

$\displaystyle y'' = -12(3x^2 + 4x - 3)$

$\displaystyle x = \frac{-4 \pm \sqrt{40}}{6} = \frac{-2 \pm \sqrt{10}}{3}$

Oh! No wonder!

I did 58x instead of 48x, wow xD

Thanks for pointing that out! :)