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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