Find the relative extrema of the function: g(x)= 2x sech x

Answer 1

The function is g(x) =2xsechx

Calculate the derivative.

Apply the product rule.

g'(x) =2x(-sechxtanhx)+2sechx

        =2sechx(-xtanhx+1)

g'(x) =0

2sechx(-xtanhx+1)=0

2sechx≠ 0

-xtanhx+1=0

xtanhx=1

Solve the equation

x =-1.2 and 1.2

Calculate the minimum value.

g(-1.2) =2(-1.2)sech(-1.2) =-0.66

Calculate the maximum value.

g(1.2) =2(1.2)sech(1.2) =0.66

Therefore the relative extreme is (-1.2,-0.66) and (1.2,0.66).