Functions f (x) and g (x) are shown below:.

f (x) f (x) = 3x2 + 12x + 16

g (x) 16g (x) = 2 sin (2x - π) + 4.

Using complete sentences, explain how to find the minimum value for each function and determine which function has the smallest minimum y-value.

First we will find the derivative of each function and equate it to zero.

f' (x) = 6 x + 12

6 x + 12 = 0

6 x = - 12

x = - 2

f ( - 2) 0 12 - 24 + 16 = 4

f (x) min = 4

g' (x) = 4 cos (2 x - π)

4 cos (2 x - π) = 0

cos (2 x - π) = 0

2 x - π = 3π / 2

2 x = 5π / 2

x = 5π/4

g (5π/4) = 2 sin (5π/2 - π) + 4 = 2 (sin 3π/2) + 4 = - 2 + 4 = 2

g (x) min = 2 (this is the smallest minimum value)