How do you find the extrema for #f(x) = x^2 +2x - 4# for [-1,1]?

1 Answer
Aug 9, 2015

Answer:

Function #f(x)=x^2+2x-4# has in #[-1,1]# a minimum #-5# for #x=-1#.

Explanation:

For any function you can use the first derivative - only a point that suffices #f'(x)=0# can be an extremum. If you're looking for extrema within a given interval you work only with those almost-extrema in the integral.

#f(x)=x^2+2x-4#
#f'(x)=2x+2#
#2x+2=0 => x=-1 in [-1,1]#

In this case the only point which can be considered in finding the extrema is #x=-1# and fortunately it's in the given interval. Now, we evaluate the second derivative and
if #f''(x)>0# then #x# is a local minimum
if #f''(x)<0# then #x# is a local maximum.

#f''(x)=2#
#f''(-1)=2>0# - #x=-1# is a local minimum and f(-1)=-5#.