Question 895132
<pre>

{{{f(theta) = 5cos(theta) + 12sin(theta)}}}

{{{"f'"(theta) = -5sin(theta)+12cos(theta)}}}

Set that = 0 to find first coordinates of possible 
maximum and minimum points:

{{{-5sin(theta)+12cos(theta)=0}}}

{{{-5sin(theta)=-12cos(theta)}}}

Divide both sides by {{{-5cos(theta)}}}

{{{(-5sin(theta))/(-5cos(theta))= (-12cos(theta))/(-5cos(theta))}}}

{{{sin(theta)/cos(theta)= 12/5}}}

{{{tan(theta) = 12/5}}}

Since tangent = {{{y/x}}} = {{{(-y)/(-x)}}}, let y=±12, x=±5 

{{{r=sqrt(x^2+y^2)}}}

{{{r=sqrt(("" +- 5)^2+ ("" +- 12)^2)}}}

{{{r=sqrt(25+144)}}}

{{{r=sqrt(169)}}}

{{{r=13}}}

{{{f(theta) = 5cos(theta) + 12sin(theta)}}}

Now we substitute both possible values to find y-coordinates
of maximum and minimum points:

{{{cos(theta) = x/r = ("" +- 5)/("" +- 13)}}} and 

{{{sin(theta) = y/r = ("" +- 12)/("" +- 13)}}}

{{{f(theta) = 5cos(theta) + 12sin(theta)}}} becomes:

{{{f(theta) = 5("" +- 5/13) + 12("" +- 12/13)}}}

{{{f(theta) = ("" +- 25/13) + ("" +- 144/13)}}}

Now we choose both possible combinations of signs:

(1)  {{{f(theta) = 25/13 + 144/13 = 169/13 = 13}}}

(2)  {{{f(theta) = -25/13 - 144/13 = -169/13 = -13}}}


(1) gives the maximum value and (2) gives the minimum value
Edwin</pre>