Find the maximum and minimum values of the following functions (a)#y= 3 sin x+ 4 cos x +2# (b) #y= 6cos x - 4 sin x -2#?

First off, we need to express these function in the form #y= acos(x+A)+b#

In this way, we can easily deduce the minimum and maximum values, by using the max and min values of #cosx#,
Which are : #1# and #-1# respectively.

#color(red)((a))#
#y=3sinx+4cosx+2#
First express #3sinx+4cosx" "# as #" "acos(x+B)#
So, you want to look for the #a, A# and #B#

#=>3sinx+4cosx=acos(x+A)#

#=>3sinx+4cosx=acosxcosA-asinxsinA#

Equate the coefficients of #sinx# and #cosx#

#=>acosA=4#

#-asinA=3=>asinA=-3#

#(asinA)/(acosA)=(-3)/4=>tanA=-3/4=>A~=-36.87º#

and

#(acosA)^2+(asinA)^2=(4)^2+(-3)^2#
#=>a^2=25=>a=5#

Hence,
#=>3sinx+4cosx=5cos(x-36.87º)#

Thus, #y=3sinx+4cosx+2=5cos(x-36.87º)+2#

The maximum value of #y# occurs when #cosx=1#
#y_"max"=5(1)+2=color(blue)(7)#

The minimum value of #y# occurs when #cosx=-1#
#y_"min"=5(-1)+2=color(blue)(-3)#

#color(red)((b))#
#y=6cosx-4sinx-2#
Similar steps as in #(a)# are repeated,

First express#6cosx-4sinx" "# as #" "acos(x+B)#
So, you want to look for the #a, A# and #B#

#=>6cosx-4sinx=acos(x+A)#

#=>6cosx-4sinx=acosxcosA-asinxsinA#

Equate the coefficients of #sinx# and #cosx#

#=>acosA=6#

#-asinA=-4=>asinA=4#

#(asinA)/(acosA)=4/6=>tanA=2/3=>A~=33.69º#

and

#(acosA)^2+(asinA)^2=(6)^2+(4)^2#
#=>a^2=52=>a=sqrt(52)#

Hence,
#=>6cosx-4sinx=sqrt(52)cos(x+33.69º)#

Thus, #y=6cosx-4sinx-2=sqrt(52)cos(x+33.69º)-2#

The maximum value of #y# occurs when #cos(x+33.69º)=1#
#y_"max"=sqrt(52)(1)-2=color(blue)(sqrt(52)-2)#

The minimum value of #y# occurs when #cos(x+33.69)=-1#
#y_"min"=sqrt(52)(-1)-2=color(blue)(-sqrt(52)-2)#