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Home » Mathematics » High School Basics (Ages 12-16)
 
Maximum of Exponential Difference
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PhireKaLk6781
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#1
Maximum of Exponential Difference
Find the value of x for which the function f(x)=2^x-4^x reaches of a maximum.

PostPosted: Tue Nov 03, 2009 5:18 pm  Back to top 
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Arrange your tan
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#2
Re: Maximum of Exponential Difference
PhireKaLk6781 wrote:
Find the value of x for which the function f(x) = 2^x - 4^x reaches of a maximum.


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Note that (2^x)^2 = 2^{2x} = (2^2)^x = 4^x


Let u = 2^x.
Then u^2 = 4^x.

f(u) = u - u^2

f(u) = - u^2 + u

Coefficients:
---------------
a = - 1
b = 1

u = \frac { - b}{2a}

u = \frac12

u = 2^x

\frac12 = 2^x

2^{ - 1} = 2^x

- 1 = x
x = - 1

f(x) = 2^x - 4^x reaches a maximum at x = - 1.






PostPosted: Tue Nov 03, 2009 6:49 pm  Back to top 
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AndrewTom
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#3
Alternatively


[Wrong forum, but we can check by]

differentiating, f(x) = 2^{x} \ln 2 - 4^{x} \ln 4 = 0

\implies \frac {4^{x}}{2^{x}} = \frac {\ln 2}{\ln 4}

\implies 2^{x} = \frac {1}{2}

\implies x = - 1 and so f(x) = \frac {1}{4}



PostPosted: Wed Nov 04, 2009 12:23 am  Back to top 
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