Question

Please help! Calculus questions

1.) Find the critical numbers of f(x)=(x-3)^1/3

2.) Can someone check my work I keep getting a really large number and it makes me think i messed up somewhere. The question is find the points of inflection and identify where the graph of f(x)=-x^4+24x^3 is concave up or concave down.
f(x)=-x^4+24x^3
f'(x)=-4x^3+72x^2
f"(x)=-12x^2+144x=0
-12x^+144x=0
12x(-x+12)=0
x=0 and x=12 are the critical #'s
so i tested -1,1,11,13 f"
f"(-1)=-156, f"(1)=132, f"(11)=132, f"(13)=-156
then used the original f(x) plugged in 0 and 12 and got (0,0) and (12,20736)

Answer 1

\[f(x)=(x-3)^{1/3}~~\Rightarrow~~f'(x)=\frac{1}{3}(x-3)^{-2/3}\]
Critical points occur for \(x\) that make \(f'(x)\) zero or undefined, in this case \(x=3\).

Answer 2

For the second question: inflection points are like critical points for a function's derivative.
\[\begin{align*}f(x)&=-x^4+24x^3\\
f'(x)&=-4x^3+72x^2\\
f''(x)&=-12x^2+144x
\end{align*}\]
Possible inflection points occurs when \(f''(x)\) is zero or undefined. Since \(f''(x)\) is a polynomial, you don't have to worry about the undefined case.
\[\begin{align*}-12x^2+144x&=0\\
x^2-12x&=0\\
x(x-12)&=0\\
x&=0,12
\end{align*}\]
Now you check the intervals:
\[\begin{matrix}
\text{interval}&&\text{test point}&&\text{sign of }f''(\text{test point})\\
(-\infty,0)&&-1&&-\\
(0,12)&&1&&+\\
(12,\infty)&&13&&-
\end{matrix}~~\iff~~\begin{matrix}.\\\text{concave down}\\\text{concave up}\\\text{concave down}\end{matrix}\]
Notice that you only need one test point from each interval - the sign of the derivative will be the same to the right of 0 as it is to the left of 12.

So you have actual inflection points at \(x=0\) and \(x=12\), which give \((0,0)\) and \((12,20736)\).

Your work is correct. Don't let big numbers make you doubt yourself. It makes sense that you get such a big number, given that you have a quartic polynomial. You should expect big numbers the further you get from the original polynomial's roots (which were \(x=0\) and \(x=24\)).

Answer 3

Thanks!

Answer 4

You're welcome!