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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: 12,184 Thanks: 481 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 rootfinding 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 subforum. 
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: 12,184 Thanks: 481 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.


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exponential, extrema, function, rational 
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