If f (x) = - x ^ 2 + 2aX and G (x) = (a + 1) ^ (1-x) are decreasing functions in the interval [1,2], find a

If f (x) = - x ^ 2 + 2aX and G (x) = (a + 1) ^ (1-x) are decreasing functions in the interval [1,2], find a

1）
If f (x) is a quadratic function, the opening is downward and the interval [1,2] is a decreasing function, then the symmetric axis X = a 1 is an increasing function
Therefore, G (x) = (a + 1) ^ (1-x) is a decreasing function when (a + 1) > 1
That is, a > 0
0 is obtained from 1) 2)

Given the function f (x) = ax ˇ 2-2ac + 2 + B (a > 0), if f (x) has a maximum value of 5 and a minimum value of 2 in the interval [2,3], (1) find the value of a and B;

There are two sets of solutions,
a=1,b=0
perhaps
a=-1,b=3

Given the function f (x) = - 3x ^ 2-6x + 1, we can find its monotone interval, and find its maximum and minimum on [- 3,0]

F (x) = - 3x ^ 2-6x + 1 = 3 (x-1) ^ 2-2, so it decreases monotonically on (- ∞, 1) and increases monotonically on [1, + ∞)
f(-3)=3（-3-1）^2-2=46
f(0)=3（0-1）^2-2=1

Given that f (x) = 2x3-6x2 + m (M is a constant) has a maximum value of 3 on [- 2,2], then the minimum value of this function on [- 2,2] is ()
A. - 37B. - 29c. - 5D. None of the above is true

∵ f ′ (x) = 6x2-12x = 6x (X-2), ∵ f (x) is an increasing function on (- 2,0) and a decreasing function on (0,2), ∵ when x = 0, f (x) = m is the maximum, ∵ M = 3, so f (- 2) = - 37, f (2) = - 5. ∵ the minimum is - 37