Exponential function in senior one Let 0 ≤ x ≤ 2, find the maximum and minimum of the function x-1/2 x y=4 - a·2 + a*a/2 + 1 ..................x-1/2 x y=4 - a·2 + a*a/2 + 1

Exponential function in senior one Let 0 ≤ x ≤ 2, find the maximum and minimum of the function x-1/2 x y=4 - a·2 + a*a/2 + 1 ..................x-1/2 x y=4 - a·2 + a*a/2 + 1

4^(x-1/2)=2^(2x-1)=(2^2x)/2=(2^x)²/2
∴y=4^(x-1/2)-a*2^x+a²/2+1
=(2^x)²/2-a*2^x+a²/2+1
Let t = 2 ^ x ∈ [1,4]
Then y = T & sup2 / 2-At + A & sup2 / 2 + 1 = (T & sup2; - 2at + A & sup2;) / 2 + 1 = (T-A) & sup2 / 2 + 1
When a ∈ [1,4],
When t = a, y has a minimum value of 1
① When a ∈ [1,2.5] and T = 4, the maximum value is y = 9-4a + A & sup2 / 2
② When a ∈ (2.5,4], t = 1, the maximum value is y = 3 / 2-A / 2 + A & sup2 / 2
When a < 1
When t = 1, y has the minimum value y = 3 / 2-A / 2 + A & sup2 / 2
When t = 4, y has the maximum value y = 9-4a + A & sup2 / 2
When a > 4
When t = 1, y has the maximum value y = 3 / 2-A / 2 + A & sup2 / 2
When t = 4, y has the minimum value y = 9-4a + A & sup2 / 2
To sum up
When a < 1, the minimum value of Y is 3 / 2-A / 2 + A & sup2 / 2, and the maximum value is 9-4a + A & sup2 / 2
When 1 ≤ a ≤ 2.5, the minimum value of Y is 1 and the maximum value is 9-4a + A & sup2 / 2
When 2.5 < a ≤ 4, the minimum value of Y is 1 and the maximum value is 3 / 2-A / 2 + A & sup2 / 2
When a > 4, the minimum value of Y is 9-4a + A & sup2 / 2, and the maximum value is 3 / 2-A / 2 + A & sup2 / 2