What are the local extrema of #f(x)= x^3-6x^2+15#, if any?

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Answer 1 · 2015-12-03T01:16:39

#(0,15),(4,-17)#

A local extremum, or a relative minimum or maximum, will occur when the derivative of a function is #0#.

So, if we find #f'(x)#, we can set it equal to #0#.

#f'(x)=3x^2-12x#

Set it equal to #0#.

#3x^2-12x=0#

#x(3x-12)=0#

Set each part equal to #0#.

#{(x=0),(3x-12=0rarrx=4):}#

The extrema occur at #(0,15)# and #(4,-17)#.

Look at them on a graph:

graph{x^3-6x^2+15 [-42.66, 49.75, -21.7, 24.54]}

The extrema, or changes in direction, are at #(0,15)# and #(4,-17)#.