how do you find the maximum and minimum values of the function on the given interval: y=x-4x/x+1
Assuming your function to be: y = x - 4x/(x + 1),
then maxima and minima are given when/where the slope is zero.
The slope is
dy/dx = 1 – {4/(x + 1) + 4x.(-1).(x + 1)^(-2)}
dy/dx = 1 – 4/(x + 1) + 4x/(x + 1)^2
when dy/dx = 0, then
(x + 1)^2 -4(x + 1) + 4x = 0
x^2 + 2x + 1 – 4x – 4 + 4x = 0
x^2 + 2x – 3 = 0
(x + 3)(x – 1) = 0
x = 1, -3
So, the maxima and minima of the function, y = x - 4x/(x + 1), are found when x = 1 and when x = -3
x = 1
y = 1 – 4/(1 + 1) = 1 – 4/2 = -1
Minimum value is at (1,-1) and equals -1.
x = -3
y = -3 + 12/(-3 + 1) = -3 – 12/2 = -9
Maximum value is at (-3,-9) and equals -9.