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September 21st, 2011, 01:22 PM   #1
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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:

?
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September 21st, 2011, 03:41 PM   #2
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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.
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September 22nd, 2011, 10:46 AM   #3
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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?
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September 22nd, 2011, 11:45 AM   #4
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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).
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September 22nd, 2011, 12:28 PM   #5
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Re: Rational function, exponential function, extrema

Thanks.
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