# Looking for inflection points

• Nov 11th 2012, 10:27 PM
camjenson
Looking for inflection points
Find the first positive x value of x for which f(x)=sin(e^x) has an inflection point. Correctly take derivatives.
• Nov 11th 2012, 10:36 PM
MarkFL
Re: Looking for inflection points
Have you found the second derivative?
• Nov 11th 2012, 10:38 PM
camjenson
Re: Looking for inflection points
Yes, it's e^x(cos(e^x)-e^xsin(e^x))
• Nov 11th 2012, 11:02 PM
MarkFL
Re: Looking for inflection points
Correct. Since $\displaystyle 0<e^x$ for all real $\displaystyle x$, this leaves you to solve:

$\displaystyle f(x)=\cos(e^x)-e^x\sin(e^x)=0$

I recommend a numeric root-finding technique, such as Newton's method. Graph the function to determine a good initial guess.
• Nov 11th 2012, 11:08 PM
camjenson
Re: Looking for inflection points
I'm sorry. I don't know Newton's method?
• Nov 11th 2012, 11:11 PM
Soroban
Re: Looking for inflection points
Hello, camjenson!

Quote:

Find the first positive value of $\displaystyle x$ for which $\displaystyle f(x)\:=\:\sin(e^x)$ has an inflection point.

We have: .$\displaystyle f(x) \:=\:\sin(e^x)$

Then: .$\displaystyle f'(x) \:=\:e^x\cdot\cos(e^x)$

Hence: .$\displaystyle f''(x) \;=\; e^x\!\cdot\!\cos(e^x) + e^x\!\cdot\![-\sin(e^x)]\!\cdot\! e^x \;=\;e^x\big[\cos(e^x) - e^x\sin(e^x)\big]$

And we must solve: .$\displaystyle \begin{Bmatrix}e^x \:=\:0 & [1] \\ \cos(e^x) - e^x\sin(e^x) \:=\:0 & [2] \end{Bmatrix}$

Equation [1] has no solutions.

Equation [2] can be approximated.

Let $\displaystyle u = e^x$
We have: .$\displaystyle \cos u - u\sin u \:=\:0 \quad\Rightarrow\quad u\sin u \:=\:\cos u \quad\Rightarrow\quad \frac{\sin u}{\cos u} \:=\:\frac{1}{u}$
And we can approximate a root of: .$\displaystyle \tan u \:=\:\frac{1}{u}$
Then back-substitute: .$\displaystyle x \,=\,\ln(u)$

• Nov 11th 2012, 11:17 PM
camjenson
Re: Looking for inflection points
WHat do you mean by back-substitute x=ln(u)? Sorry I'm a little slow...
• Nov 11th 2012, 11:20 PM
MarkFL
Re: Looking for inflection points
Quote:

Originally Posted by Soroban
...

The OP has the same result that you cite. :)

Quote:

Originally Posted by camjenson
I'm sorry. I don't know Newton's method?

You should be able to find it in your textbook, or online. I assume you are expected to know it, since you have been given this problem.
• Nov 12th 2012, 10:05 PM
hollywood
Re: Looking for inflection points
Or perhaps the professor expects you to use a graphing calculator to find the root. The calculator uses Newton's method, but you don't need to know that to plug in the functions and get your answer.

- Hollywood