Find the maximum and minimum values of the function y = x / 1 + X & # 178 on the closed interval [- 1,1]

Find the maximum and minimum values of the function y = x / 1 + X & # 178 on the closed interval [- 1,1]

Let X1 and X2 belong to (- 1,1), where X1 < x2
Then f (x1) - f (x2)
=x1/(1+x1^2)-x2/(1+x2^2)
=[x1（1+x2^2）-x2（1+x1^2）]/(1+x1^2)(1+x2^2)
=[(x1-x2)+x1x2(x2-x1）]/(1+x1^2)(1+x2^2)
=(x1-x2)(1-x1x2)/(1+x1^2)(1+x2^2)
X1, X2 belong to [- 1,1],
Know x1x2 < 1
That is 1-x1x2 > 0
From x1-x2 < 0
Know (x1-x2) (1-x1x2) / (1 + X1 ^ 2) (1 + x2 ^ 2) < 0
That is, f (x1) - f (x2) < 0
That is, f (x) is an increasing function on [- 1,1],
So when x = 1, y has the maximum f (1) = 1 / 2
When x = - 1, y has the minimum value f (- 1) = - 1 / 2

Find the maximum and minimum values of the function y = (1 / 2) ˊ x2-6x + 17 in the interval [2,4]

The maximum and minimum values of the function y = (1 / 2) ^ (x ^ 2-6x + 17) in the interval [2,4] x ^ 2-6x + 17 = (x-3) ^ 2 + 8 when x ∈ [2,4], the minimum value of (x-3) ^ 2 + 8 is 8 (when x = 3), because the original function is an exponential function with the base of 1 / 2 and is decreasing, so the maximum value of the original function is (1 / 2) ^ 8 = 1

The function y = - X & sup2; + 6x + 9 has a maximum value of 9 and a minimum value of - 7 in the interval [a, b] (a < B < 3). Find the value of real numbers a and B

The axis of symmetry is x = 3
So it is monotonically increasing from negative infinity to 3
When x = a, y = - 7
When x = B, y = 9
When x = a, y = - 7, a = 8 or - 2, a = 8 is rounded off
When x = B, y = 9, B = 0 or 6, B = 6 are rounded off
In conclusion, a = - 2, B = 0