Question

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.

Answer 1

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 )