 My Math Forum Rational function, exponential function, extrema
 User Name Remember Me? Password

 Calculus Calculus Math Forum

 September 21st, 2011, 01:22 PM #1 Newbie   Joined: Sep 2011 Posts: 18 Thanks: 0 Rational function, exponential function, extrema Hi! Could someone explain to me, step by step, how to find maximum and minimum of a function like this: ? September 21st, 2011, 03:41 PM #2 Senior Member   Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Rational function, exponential function, extrema We are given: To find the extrema, we look for values where f'(x) = 0 or undefined: Using the quotient and power rules for differentiation, we obtain: We see that the denominator has no real roots, thus we need only simplify the numerator and find its roots: Apply quadratic formula: These two roots will be where f(x) has its possible extrema. Dividing the number line at these critical values, and checking the sign of f'(x) in the resulting intervals shows: Interval : f'(-3) = (+)/(+) = + f(x) increasing on this interval. Interval f'(-1) = (-)/(+) = - f(x) decreasing on this interval. Interval f'(0) = (+)/(+) = + f(x) increasing on this interval. So, we find a local maximum at Thus, the maximum is at the point Since , we know this maximum is a global maximum. See if you can now find the xy coordinates of the minimum at the other critical value, and identify it as a global minimum. One can also use f''(x) to identify the type of extrema using the concavity of f(x). We may compute: Using a root-finding technique, we find f''(x) has roots at: As there are no roots of multiplicity, and f''(0) > 0, we conclude for f(x): Concave up on (-?,-4.07123) U (-0.754374,0.325603) Concave down on (-4.07123,-0.754374) U (0.325603,?) We find f''(x) is negative at the smaller critical value (?-2.8228 , indicating a maximum. You can verify that f''(x) is positive at the other critical value (?-0.177124) indicating a minimum. This is a topic in elementary calculus, so I have moved it to a more appropriate sub-forum. September 22nd, 2011, 10:46 AM #3 Newbie   Joined: Sep 2011 Posts: 18 Thanks: 0 Re: Rational function, exponential function, extrema Thank you very much. I tried using it for x=2^n and it doesn't work because x must be positive then. Could you tell me what changes if x=a^n? September 22nd, 2011, 11:45 AM #4 Senior Member   Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Rational function, exponential function, extrema If then we may implicitly differentiate with respect to n by first converting from exponential to logarithmic form: Note: Since there are no extrema. You only have: global minimum of 0 and global maximum of �? (same sign as k). September 22nd, 2011, 12:28 PM #5 Newbie   Joined: Sep 2011 Posts: 18 Thanks: 0 Re: Rational function, exponential function, extrema Thanks. Tags exponential, extrema, function, rational ,

,

,

,

,

,

,

,

,

,

,

,

,

,

# maximum value of a rational function

Click on a term to search for related topics.
 Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post jones1234 Calculus 6 January 5th, 2013 01:48 PM maxgeo Algebra 6 August 28th, 2012 08:02 PM goliath Algebra 2 October 15th, 2009 03:02 PM axelle Algebra 1 October 13th, 2007 05:22 PM arnold Real Analysis 1 December 31st, 1969 04:00 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top      