"compare the functions shown below:

f(x) = 3 cos 2x + 4
g(x)

cosine graph with points at 0, 3 and pi over 2, 0 and pi, negative 3 and 3 pi over 2, 0 and 2 pi, 3
h(x)

xy
-6-11
-5-6
-4-3
-3-2
-2-3
-1-6
0-11"

We have to calculate the maximum and minimum of these functions.

f(x)=3 cos (2x)+4

1) we find the first derivative
f´(x)=-6 sin(2x)

2) We find those values that makes the first derivative equal to zero.
-6 sin(2x)=0
sin (2x)=0/(-6)
sin (2x)=0
2x=sin⁻¹ 0
2x=kπ
x=kπ/2               K=(...,-2,-1,0,1,2,...)

2) we find the second derivative and check if it has a maximum or minimum at x=kπ/2
f´´(x)=-12 cos (2x)
for example if k=0;
f´´(0)=-12 cos(2*0)=-12<0 ; because -12 is less than "0" ,it  has a maximum at x=kπ/2.

3) we find the maximum y-value:
if K=0; ⇒x=0

f(x)=3 cos (2x)+4
f(0)=3 cos (2*0)+4=3+4=7

The maximum y-value of f(x)=3 cos (2x)+4 is y=7.

g(x)
We can look at the graph of this function :
the maximum y-value is y=3.

h(x)
We can look at the table of this function;
 the maximum y-value of this function is y=-2

Therefore the greatest maximum y-value will be y=7

Which function has the greatest maximum y-value?
f(x)

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