How do I find the extrema of a function?

Question

Anna Allen · Accepted Answer

Check below.

Given a point #M(x_0,f(x_0))#, if #f# is decreasing in #[a,x_0]# and increasing in #[x_0,b]# then we say #f# has a local minimum at #x_0#, #f(x_0)=...# If #f# is increasing in #[a,x_0]# and decreasing in #[x_0,b]# then we say #f# has a local maximum at #x_0#, #f(x_0)=....# More specifically, given #f# with domain #A# we say that #f# has a local maximum at #x_0##in##A# when there is #δ>0# for which #f(x)<=f(x_0)# , #x##inAnn##(x_0-δ,x_0+δ)# , In similar way, local min when #f(x)>=f(x_0)# If #f(x)<=f(x_0)# or #f(x)>=f(x_0)# is true for ALL #x##in##A# then #f# has an extrema (absolute) If #f# has no other local extremas in its domain #D_f# then we say #f# has an extrema (absolute) at #x_0#. Creating a monotony table in each case where you can study #f'# sign and #f# monotony in their domain will make things easier.

Xavier Alonso · Answer

undefined To find the extrema of a function, you need to follow these steps:

1. Find the critical points of the function by setting its derivative equal to zero and solving for x.
2. Determine the value of the function at each critical point.
3. Evaluate the function at the endpoints of the domain, if applicable.
4. Compare the values obtained in steps 2 and 3 to determine the maximum and minimum values of the function.