My Math Forum Definite Integrals, extrema

 Calculus Calculus Math Forum

 May 8th, 2009, 05:39 PM #1 Joined: May 2009 Posts: 58 Thanks: 0 Definite Integrals, extrema Find and classify the relative maxima and minima of f(x), if f(x) = integral from 0 to x of {(t^2-4)dt} /{1+(cost)^2} I got x=2 and x=-2 for f'(x)=0, but, I don't know how to calculate the relative maxima and minima. Please help me. Thank you
 May 9th, 2009, 09:07 AM #2 Joined: Dec 2008 Posts: 251 Thanks: 0 Re: Definite Integrals, extrema Here, we may use the fact that $\frac{d}{dx}\int\,_0^x\,g(t)\,dt\,=\,g(x).$ (As you move $x$ to the right, the rate of change of the area under a curve is equal to the height of the function.) Your solutions $x\,=\,\pm 2$ are correct. To prove that they are relative maxima and minima, we can use the Second Derivative Test: $\begin{eqnarray*} f''(x) &>& 0\mbox{ }\rightarrow\mbox{ }x\mbox{ is a local minimum}\\ f''(x) &<& 0\mbox{ }\rightarrow\mbox{ }x\mbox{ is a local maximum} \end{eqnarray*}$
 May 10th, 2009, 04:53 AM #3 Joined: May 2009 Posts: 58 Thanks: 0 Re: Definite Integrals, extrema Thanks for your help. But, how can I calculate f(2)? is x=-2 also a solution since the domain of f(x) is x>=0 ?
 May 10th, 2009, 05:23 AM #4 Joined: Dec 2008 Posts: 251 Thanks: 0 Re: Definite Integrals, extrema If the domain is, as you say, $[0,\,\infty)$, then $x\,=\,2$ is the only local extremum. However, since $\cos\,x$ is squared and added to $1$ in the denominator, $f(x)$ is defined everywhere. I just put the integral into the Wolfram Online Integrator and it gave an answer that was not expressed in terms of elementary functions. If the problem only asks to classify the extrema, then I don't think you have to worry about the values of the extrema at those points.

 Tags definite, extrema, integrals

### indefinite integral finding relative maxima and minima

Click on a term to search for related topics.
 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post Agata78 Calculus 6 January 19th, 2013 03:05 PM Agata78 Calculus 18 January 18th, 2013 01:39 PM jakeward123 Calculus 10 February 28th, 2011 01:18 PM Aurica Calculus 2 May 10th, 2009 05:05 PM Agata78 Abstract Algebra 0 January 1st, 1970 12:00 AM

 Contact - Home - Top