In your lab, a substance's temperature has been observed to follow the function T(x) = (x − 4)3 + 6. The turning point of the graph is where the substance changes from a liquid to a gas. Using complete sentences in your written answer, explain to your fellow scientists how to find the turning point of this function. Hint: The turning point of the graph is similar to the vertex of a quadratic function.

Question

alekos · Answer

That would be where the derivative equals zero

flvskidd · Answer

is that my answer?

anonymous · Answer

Okay. I think the a,b, and c  are from
a x^3 + b x^2 +c x + d

unfortunately, you have   (x - 4)^3 + 6
which is  (x-4) (x-4) (x-4) + 6
we have to multiply that out to get it in the form so we can find a, b and c

anonymous · Answer

first do (x-4)(x-4)

anonymous · Answer

Which is

$$x^2 - 8x + 16$$

anonymous · Answer

Now do:

$$(x^2 -8x+16)(x-4)$$

anonymous · Answer

Which is:

$$x^3 - 12x^2 + 48x - 64$$

anonymous · Answer

So your function is:

$$(x−4)^3+6=x^3−12x^2+48x−64+6=x^3−12x^2+48x−58$$

anonymous · Answer

so A=1 B=-12 C=48

anonymous · Answer

So it the answer is x=12/3 = 4
that is the turning point of this curve
Here is a graph:

anonymous · Answer

Hope this helped! Have a great day!

Also a medal would be much appreciated! Just click best response next to my answer. Thank You!

@FLVSKidd

flvskidd · Answer

Thank you Tom. Yes it did help.

anonymous · Answer

Haha. No problem!

@FLVSKidd

alekos · Answer

that's one way of doing it but...........

taking the derivative of the function is much easier.
then make it equal to zero to get the turning point