How could I find the maxim and the minim values, if they exist, for a function like f(x)=3^(sin x)
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In the function f(x) = 3^(sin x) the variable part is sinx. This has a maximum value of 1 and minimum value of -1. Further 3 is a positive number greater than 1, the value of 3^(sinx) is maximum when sinx is maximum, and minimum when sin x is minimum.
Therefore maximum of 3^(sin x) = 3^1 = 3
and minimum of 3^(sin x) = 3^(-1) = 1/3
Please not that when sinx is equal to 0, the value of 3^(sinx) is equal to 1.
As a general rule we can say that for a function of the type a^[f(x)]:
If a >1, the function is maximum when f(x) is maximum and minimum when f(x) is minimum.
If 0 < a < 1, the function is minimum when f(x) is maximum and maximum when f(x) is minimum.
If a = 1, the function is always equal to 1.
When a is negative the function fill fluctuate between negative and positive values with increasing value of f(x).
By common sense, f(x) = 3^sinx has the highest value when sinx is maximum and f(x) is minimum when sinx is minimum.
sinx = 1, maximum when x = 90 degree. or pi/2 rad.
The value of f(x) = 3^sin(pi/2) = 3.
sinx = -1, mimimum when x = 270 degree or x = 3pi/2 .
So the mimum value of f(x) = 3^sin(3pi/2) = 3^(-1) = 1/3.
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