Another Question...

[ChubbyBoi]

9. For what values of x does the curve y = -x^3 + 6x^2 have a slope of -12? Of -15? Use a graphing calculator to graph the function and confirm your results.

Is it correct to use the quadratic formula for this derivative of this? I am not getting the right answer of 2+- 2sqroot3

Help would be much appreciated.

 

[JeffM]

9. For what values of x does the curve y = -x^3 + 6x^2 have a slope of -12? Of -15? Use a graphing calculator to graph the function and confirm your results.

Is it correct to use the quadratic formula for this derivative of this? I am not getting the right answer of 2+- 2sqroot3

Help would be much appreciated.

Yes, it is correct to use the quadratic formula.

You are asked two questions. What is your derivative? What are you getting as the results of your two quadratic formulas?

Hard to know where you are going wrong if you show no work.

 

[ChubbyBoi]

Yes, it is correct to use the quadratic formula.

You are asked two questions. What is your derivative? What are you getting as the results of your two quadratic formulas?

Hard to know where you are going wrong if you show no work.

My apologies JeffM, my intentions were to clarify how to start the solving the problem as I was unsure. Here is my work for the first half of the question, as the second part is similar, I can just use the same method:

 

[JeffM]

My apologies JeffM, my intentions were to clarify how to start the solving the problem as I was unsure. Here is my work for the first half of the question, as the second part is similar, I can just use the same method:

You found the derivative properly

\(\displaystyle y = -x^3 + 6x^2 \implies y' = - 3x^2 + 12x.\)

\(\displaystyle y' = - 12 \implies -12 = -3x^2 + 12x \implies -4 = - x^2 + 4x \implies x^2 - 4x - 4 = 0 \implies x = \dfrac{-(- 4) \pm \sqrt{(-4^2) - 4 * (1)(-4)}}{2 * 1} \implies\)

\(\displaystyle x = \dfrac{4 \pm \sqrt{16 + 16}}{2} = \dfrac{4 \pm \sqrt{32}}{2} = \dfrac{4 \pm \sqrt{16 * 2}}{2} = \dfrac{4 \pm 4\sqrt{2}}{2} = 2 \pm 2\sqrt{2}.\)

Once again, I do not agree with your answer key.

\(\displaystyle x = 2 + 2\sqrt{3} \implies - 3x^2 = - 3(4 + 8\sqrt{3} + 4 * 3) = -3(16 + 8\sqrt{3}) = - 48 - 24\sqrt{3}\ and\ 12x = 24 + 24\sqrt{3}\implies\)

\(\displaystyle y' = - 48 - 24\sqrt{3} + 24 + 24\sqrt{3} = - 24 \ne - 12.\)

But \(\displaystyle x = 2 + 2\sqrt{2} \implies - 3x^2 = - 3(4 + 8\sqrt{2} + 4 * 2) = -3(12 + 8\sqrt{2}) = - 36 - 24\sqrt{2}\ and\ 12x = 24 + 24\sqrt{2}\implies\)

\(\displaystyle y' = - 36 - 24\sqrt{2} + 24 + 24\sqrt{2} = - 12.\)

Whence are you getting these answers?

 

[ChubbyBoi]

You found the derivative properly

\(\displaystyle y = -x^3 + 6x^2 \implies y' = - 3x^2 + 12x.\)

\(\displaystyle y' = - 12 \implies -12 = -3x^2 + 12x \implies -4 = - x^2 + 4x \implies x^2 - 4x - 4 = 0 \implies x = \dfrac{-(- 4) \pm \sqrt{(-4^2) - 4 * (1)(-4)}}{2 * 1} \implies\)

\(\displaystyle x = \dfrac{4 \pm \sqrt{16 + 16}}{2} = \dfrac{4 \pm \sqrt{32}}{2} = \dfrac{4 \pm \sqrt{16 * 2}}{2} = \dfrac{4 \pm 4\sqrt{2}}{2} = 2 \pm 2\sqrt{2}.\)

Once again, I do not agree with your answer key.

\(\displaystyle x = 2 + 2\sqrt{3} \implies - 3x^2 = - 3(4 + 8\sqrt{3} + 4 * 3) = -3(16 + 8\sqrt{3}) = - 48 - 24\sqrt{3}\ and\ 12x = 24 + 24\sqrt{3}\implies\)

\(\displaystyle y' = - 48 - 24\sqrt{3} + 24 + 24\sqrt{3} = - 24 \ne - 12.\)

But \(\displaystyle x = 2 + 2\sqrt{2} \implies - 3x^2 = - 3(4 + 8\sqrt{3} + 4 * 2) = -3(12 + 8\sqrt{2}) = - 36 - 24\sqrt{2}\ and\ 12x = 24 + 24\sqrt{2}\implies\)

\(\displaystyle y' = - 36 - 24\sqrt{2} + 24 + 24\sqrt{3} = - 24 = - 12.\)

Whence are you getting these answers?

Yes, this makes sense to me. I guess there are a few typos in the answer key and this happens to be one of them.
Just curious, why did you type = -24 = -12 in the last line, I think you meant + 24\sqrt{2} as well.
Also, is there an application you use to type out these lines?

Thanks for helping.

 

[JeffM]

Yes, this makes sense to me. I guess there are a few typos in the answer key and this happens to be one of them.
Just curious, why did you type = -24 = -12 in the last line, I think you meant + 24\sqrt{2} as well.
Also, is there an application you use to type out these lines?

Thanks for helping.

Because like the answer key I made a mistake.