Determine whether each function has a maximum or minimum value. then find the value.

y = -x^2 + 4x - 4
a. maximum; 0
b. maximum; 4
c. minimum; 0
d. minimum; 4

y = 0.5x^2 + 8x + 5
a. maximum; 19
b. maximum; 8
c. minimum; 2
d. minimum; -27

in advance!

Y=-x^2+4x-4

dy/dx=-2x+4 and d2y/dx2=-2

Since acceleration, d2y/dx2, is a constant negative, when velocity, dy/dx=0, it will be at the point when y is at an absolute maximum...

dy/dx=0 only when -2x+4=0, 2x=4, x=2, and the maximum is y(2), in this case:

y(2)=-x^2+4x-4

y(2)=-4+8-4=0

The absolute maximum is 0.

...

y=0.5x^2+8x+5

dy/dx=x+8, d2y/dx2=1

Acceleration is a constant positive, so when dy/dx=0, y(x) will be at an absolute minimum...

dy/dx=0 only when x+8=0, x=-8, then the absolute minimum will be at y(-8):

y=0.5x^2+8x+5 

y(-8)=32-64+5=-27

So the minimum value for y is -27