Find the absolute extrema of the function on a closed interval

Question

g(x) = 7 secx,  [-pi/6, pi/3], solving for minimum (x,y) and maximum (x,y).

Answer 1

g(x) is continuous on the given interval.  So, the absolute extrema exist and must occur either at a critical point or endpoint.

 

g'(x) = 7secxtanx

 

Critical points when g'(x) = 0 or g'(x) is undefined

 

       g'(x) = 0 when secx = 0 or tanx = 0

 

       g'(x) is defined for all x in the given interval

 

       secx is nonzero on the given interval

 

       tanx = 0 on the given interval only when x = 0 

 

So, the only critical point on the interval occurs when x = 0.

 

Evaluate g(x) at 0 and at each endpoint:

 

       g(0) = 7sec(0)tan(0) = 0

 

       g(-π/6) = 7(2/√3)(-1/√3) = -14/3 ← absolute minimum value

 

       g(π/3) = 7(2)(√3) = 14√3 ← absolute maximum value

 

absolute minimum point:  (-π/6, -14/3)

 

absolute maximum point:   (π/3, 14√3)