First, you need to differentiate your equation
#f(x)=3e^(-2x^2)#
#f'(x)=-4xtimes3e^(-2x^2)#
#f'(x)=-12xe^(-2x^2)#
#f''(x)=-12xtimes-4xe^(-2x^2)+e^(-2x^2)times-12#
#f''(x)=48x^2e^(-2x^2)-12e^(-2x^2)#
For stationary points/critical points, #f'(x)=0#
#-12xe^(-2x^2)=0#
#-12x=0# or #e^(-2x^2)=0#
There is no solution for #e^(-2x^2)=0# since the graph NEVER goes to 0. It only APPROACHES 0.
#-12x=0#
#x=0#
To find whether it is maximum or minimum, you sub #x=0# into #f''(x)#. If the answer is greater than zero ie #>0#, then it is a minimum. If the answer is smaller than zero ie #<0#, then it is a maximum.
#f''(0)=-12e^0 = -12 <0#
Therefore, at #x=0#, it is a maximum
To find the coordinate, sub #x=0# back into #f(x)# and you will get #y=3# --> #(0,3)# is a maximum
