What are the critical points of #f(x) =xsqrt(e^x-3x)#?

1 Answer
Feb 8, 2016

Answer:

They are approximately #0.619#, #1.512#, and #0.395#.

Explanation:

#f'(x) = (xe^x-2e^x-9x)/(2sqrt(e^x-3x)#

The critical numbers of #f# are the solutions to

#e^x-3x=0#

and the solution to #xe^x-2e^x-9x=0# that gives #e^x-3x >= 0# (so that it is in the domain of #f#.

Use whatever numerical/technological methods you have to get approximations.

#0.619# and #1.512# solve the first equation and

#0.395# and #1.234# solve the second, but #e^1.234-3(1.234) < 0# so #1.234# is not in the domain of #f# and hence is not a critical number for #f#.