Find the maximum value of the function y = Log1 / 3 ^ (x ^ 2-6x + 10) in the interval [1,2]? The key point is how to find the best value of composite function?

Find the maximum value of the function y = Log1 / 3 ^ (x ^ 2-6x + 10) in the interval [1,2]? The key point is how to find the best value of composite function?

This is a composite function; the main function Log1 / 3 G (x) (the base of the logarithmic function of the main function Log1 / 3 is a decreasing function)
True number G (x) = x ^ 2-6x + 10 (according to △ 0, according to image)
(- infinity, 3) g (x) is a decreasing function;
So in [1,2]; G (x) is a decreasing function
For compound function (same increase different decrease)
In [1,2]
y=log1/3^(x^2-6x+10)
It is an increasing function;
Ymax=y（2）=log1/3（2）
Can it solve your problem?

Monotone increasing interval and monotone decreasing interval of function Log1 / 2 ^ x ^ 2-6x + 8

X ^ 2-6x + 8 monotone increasing interval [3, positive infinity]
Monotone decreasing interval (4, positive infinity) of function Log1 / 2 ^ x ^ 2-6x + 8
Monotone increasing interval of function Log1 / 2 ^ x ^ 2-6x + 8 (negative infinity, 2)
This is because x ^ 2-6x + 8 > 0

Y = arctan (2 + 3 ^ x) inverse function

tany=tanarctan(2+3^x)=2+x^3
X ^ 3 = tany-2 x = cubic root (tany-2)
The inverse function is y = cubic root (tanx-2)