If the function f (x) is monotone in [M, n], then the difference between the maximum and minimum value of function f (x) in [M, n] is ()

If the function f (x) is monotone in [M, n], then the difference between the maximum and minimum value of function f (x) in [M, n] is ()

If monotonically increasing, then: the minimum value is f (m), the maximum value is f (n), so the difference is f (n) - f (m)
If monotonically decreasing, then: the minimum value is f (n), the maximum value is f (m), so the difference is f (m) - f (n)

Find the maximum and minimum value of function f (x) = x + A / x + 1 when x ∈ [1,2]

y=(x+a)/(x+1)
yx+y=x+a
(y-1)x=a-y
x=(a-y)/(y-1)
1≤(a-y)/(y-1)≤2
y> At 1:00,
y-1≤a-y≤2y-2
y-1≤a-y,a-y≤2y-2
1＜(a+2)/3≤y≤(a+1)/2
1＜(a+2)/3≤(a+1)/2
a＞1,
When a > 1, 1 ＜ (a + 2) / 3 ≤ y ≤ (a + 1) / 2, there is a minimum value (a + 2) / 3 and a maximum value (a + 1) / 2;
When y < 1,
2y-2≤a-y≤y-1
(a+1)/2≤y≤(a+2)/3＜1
(a+1)/2≤(a+2)/3＜1
a＜1
When a ＜ 1, (a + 1) / 2 ≤ y ＜ (a + 2) / 3 ＜ 1, there is a minimum (a + 1) / 2 and a maximum (a + 2) / 3;
When f = 1
in summary
When a > 1, f (x) has a minimum value (a + 2) / 3 and a maximum value (a + 1) / 2;
When a < 1, f (x) has a minimum value (a + 1) / 2 and a maximum value (a + 2) / 3;
When a = 1, f (x) = 1

If f (x) = a ^ x (a ＞ 0 and a ≠ 1), on [1,2], the maximum value is greater than the minimum value by a / 2, then a?

a> 1, f (x) = a ^ x is an increasing function, a & sup2; - a = A / 2, a = 3 / 2
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