# Thread: [SOLVED] curve sketching problem

1. ## [SOLVED] curve sketching problem

Consider the function $\displaystyle f(x) = e^{arctan(2x)}$

1. Find the horizontal asymptotes of the function.
2. State the interval(s) where the function is increasing.
3. Find the inflection point.
4. Find the intervals where the graph is concave up and concave down.

It appears I've ran into a problem here. All I could do on my own so far was find f'(x):
$\displaystyle f'(x) = e^{arctan(2x)}(\frac{2}{1+4x^2}) = \frac{2e^{arctan(2x)}}{1+4x^2}$

Setting this equal to 0, $\displaystyle e^{arctan(2x)} = 0$ which has no solution? Doesn't look quite right to me.

2. That's correct -- the function is monotonic over $\displaystyle \mathbb{R}$

3. Ah, thank you! I should have noticed that just by looking at the function itself.

I'm still not completely positive on whether or not I have done the rest of the problem correctly.
This is what I have for the second derivative:

$\displaystyle f''(x) = \frac{(1+4x^2)\frac{4e^{arctan(2x)}}{(1+4x^2)} - (2e^{arctan(2x)})(8x)}{(1+4x^2)^2}$

Which then simplifies to be:

$\displaystyle f''(x) = (\frac{2e^{arctan(2x)}}{(1+4x^2)^2})(2-8x)$

I suppose the only way for it to equal zero would be x=1/4
So the coordinates of the point of inflection are $\displaystyle (\frac{1}{4}, e^{arctan(1/2)})$

And it would be concave up on $\displaystyle (-\infty,\frac{1}{4})$ and concave down on $\displaystyle (\frac{1}{4},\infty)$

EDIT: Already submitted this answer and it turned out to be correct
The text contains a series of mathematical questions related to the function \( f(x) = e^{\arctan(2x)} \), and answers to some of these questions are provided.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Thread: [SOLVED] curve sketching problem

1. ## [SOLVED] curve sketching problem

Consider the function $\displaystyle f(x) = e^{arctan(2x)}$

1. Find the horizontal asymptotes of the function.
2. State the interval(s) where the function is increasing.
3. Find the inflection point.
4. Find the intervals where the graph is concave up and concave down.

It appears I've ran into a problem here. All I could do on my own so far was find f'(x):
$\displaystyle f'(x) = e^{arctan(2x)}(\frac{2}{1+4x^2}) = \frac{2e^{arctan(2x)}}{1+4x^2}$

Setting this equal to 0, $\displaystyle e^{arctan(2x)} = 0$ which has no solution? Doesn't look quite right to me.

2. That's correct -- the function is monotonic over $\displaystyle \mathbb{R}$

3. Ah, thank you! I should have noticed that just by looking at the function itself.

I'm still not completely positive on whether or not I have done the rest of the problem correctly.
This is what I have for the second derivative:

$\displaystyle f''(x) = \frac{(1+4x^2)\frac{4e^{arctan(2x)}}{(1+4x^2)} - (2e^{arctan(2x)})(8x)}{(1+4x^2)^2}$

Which then simplifies to be:

$\displaystyle f''(x) = (\frac{2e^{arctan(2x)}}{(1+4x^2)^2})(2-8x)$

I suppose the only way for it to equal zero would be x=1/4
So the coordinates of the point of inflection are $\displaystyle (\frac{1}{4}, e^{arctan(1/2)})$

And it would be concave up on $\displaystyle (-\infty,\frac{1}{4})$ and concave down on $\displaystyle (\frac{1}{4},\infty)$

EDIT: Already submitted this answer and it turned out to be correct
user
The text contains a series of mathematical questions related to the function \( f(x) = e^{\arctan(2x)} \), and answers to some of these questions are provided.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS