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We are just wrapping up the first semester calculus with drawing graphs of functions. I sometimes feel like my reasoning is a bit shady when I am doing that, so I decided to ask you people from Math.SE.

I am supposed to draw a graph (and show my working) of the function $f(x)=-x\sqrt{1-x^2}$: Below is my work, I'd be very grateful for any comments on possible loopholes in my reasoning (the results should be correct), thanks!


$$\text{1. } f(x)=-x\sqrt{1-x^2}$$

Domain: We have a square-root function, therefore we need $1-x^2\geq0$. From that we get$x\in[-1,1]$. As this is the only necessary condition, $D(f)=[-1,1]$. The function is also continuous on this interval as there is nothing that would produce a discontinuity.

Symmetry: $f(x)$ is an odd function, as $-f(x)=f(-x)$, as demonstrated here: $$-f(x)=f(-x)$$ $$-(-x\sqrt{1-x^2})=-(-x)\sqrt{1-(-x)^2}$$ $$x\sqrt{1-x^2}=x\sqrt{1-x^2}$$

Therefore we are only interested in the interval $[0,1]$, as the function will behave symmetrically on $[-1,0]$.

x and y intercepts: $f(x)=0$ holds when $x=0,1$, those are then the x-intercepts. Thus also the y-intercept is at $(0,0)$

First derivative $$\begin{align} \ f'(x) &=(-x\sqrt{1-x^2})' \\ & = (-x)'\sqrt{1-x^2}+(-x)(\sqrt{1-x^2})' \\ & = -\sqrt{1-x^2}+(-x)\frac{1}{2\sqrt{1-x^2}}(-2x) \\ & = -\frac{1-x^2}{\sqrt{1-x^2}}+\frac{x^2}{\sqrt{1-x^2}} \\ & = \frac{2x^2-1}{\sqrt{1-x^2}} \\ \end{align}$$

Local minima and maxima: The equality $\frac{2x^2-1}{\sqrt{1-x^2}}=0$ yields $x=\frac{1}{\sqrt2}$. As $f(\frac{1}{\sqrt2})=-\frac{1}{2}$ and because we know the x-intercepts, we can conclude that this is the local minimum and the function is decreasing at $x\in [0,\frac{1}{\sqrt2})$ and increasing at $x\in(\frac{1}{\sqrt2},1]$. That can also be concluded from the fact that $f′≤0$ on the interval $[0,\frac{1}{\sqrt2}]$ and $f′≥0$ on the interval $[\frac{1}{\sqrt2},1]$.

Second derivative $$\begin{align} \ f''(x) &=(\frac{2x^2-1}{\sqrt{1-x^2}})' \\ & = \frac{(2x^2-1)'\sqrt{1-x^2}-(2x^2-1)(\sqrt{1-x^2})'}{1-x^2} \\ & = \frac{4x\sqrt{1-x^2}-(2x^2-1)\frac{-x}{\sqrt{1-x^2}}}{1-x^2} \\ & = \frac{4x\frac{1-x^2}{\sqrt{1-x^2}}-(2x^2-1)\frac{-x}{\sqrt{1-x^2}}}{1-x^2} \\ & = \frac{\frac{-2x^3+3x}{\sqrt{1-x^2}}}{1-x^2} \\ & = \frac{-2x^3+3x}{\sqrt{(1-x^2)^3}} \\ \end{align}$$

Inflection point and concavity: $f''(x)=0$ has only one result and that is $x=0$. If we look at the inequality for $x\in(0,1]$. $$\begin{align} \ 0&<f''(x) \\ 0&<\frac{-2x^3+3x}{\sqrt{(1-x^2)^3}} \\ 0&<-2x^3+3x \\ 2x^3&<3x \\ 2x^2&<3 \\ x^2&<3/2 \\ \end{align}$$

We can see that $x^2<3/2$ is satisfied on $x\in(0,1]$, thus the function is convex on this interval.

Conclusion: Using the symmetricity of the function, we can conclude that the function is increasing on $[-1,-\frac{1}{\sqrt2}]$ and $[\frac{1}{\sqrt2},1]$ and decreasing on $[-\frac{1}{\sqrt2},\frac{1}{\sqrt2}]$. It has local minimum at $x=\frac{1}{\sqrt2}$ and local maximum at $x=-\frac{1}{\sqrt2}$. It is concave on $[-1,0]$ and convex on $[0,1]$, with point of inflection at $x=0$.

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This looks very nice. I have only two suggestions: 1) ephasize throughout that you are only considering the interval $[0,1]$. e.g, "in the interval $[0,1]$, $f''(x)=0$ has only the solution $x=0$. 2) When examining the local mins and maxes, it would be easier and more rigorous to just note $f'\le0$ on the interval $[0,1/\sqrt 2]$ and $f'\ge 0$ on the interval $[1/\sqrt2,1]$. –  David Mitra Jan 10 '13 at 17:38
    
Sounds good to me, good job ! –  Alan Simonin Jan 10 '13 at 17:39
2  
(+1) for an archetypal question –  The Chaz 2.0 Jan 10 '13 at 17:40
    
@DavidMitra Thanks for the suggestion. I've made a minor edit following your second suggestion. –  Dahn Jahn Jan 10 '13 at 17:45
    
@TheChaz2.0 Actually, regarding that, what is the best way to name this question. I am asking because a) for purposes of easy findability for other users and b) in my language, Czech, we have a fixed word for this kind of a thing, whereas English doesn't seem to have one (or am I wrong?). –  Dahn Jahn Jan 11 '13 at 1:01
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1 Answer

up vote 1 down vote accepted

Your answer and/or analysis of the function $$f(x)=-x\sqrt{1-x^2}$$ is accurate, thorough, well-justified, and consistent with its graph:


$\quad \quad f(x)=-x\sqrt{1-x^2}$.

Wolfram Alpha: $f(x)=-x\sqrt{1-x^2}$ $\quad$Source: Wolfram Alpha.


Kudos for the effort you've shown and your accurate and detailed analysis!

(Just don't forget to graph the function too!)


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2  
Haha, thanks, I've edited the question, as I am supposed to do all the working out. I am more interested in not giving claims I haven't explicitly proven (or at least demonstrated enough), rather than correct answers (which I can, as you pointed out, check with software/wolfram alpha) –  Dahn Jahn Jan 10 '13 at 17:37
    
Absolutely - I like to back up claims as well, and your analysis is more accurate than simply trying to "eye-up"/estimating the intervals on which it is increasing, decreasing, etc. etc. Nice job with your assessment/analysis. –  amWhy Jan 10 '13 at 17:40
    
Since it seems like Math.SE approves, I chose this as the accepted question, no point waiting for anything else. –  Dahn Jahn Jan 10 '13 at 18:34
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