How do you determine whether the function #f(x)=x^(4)-6x^(3)# is concave up or concave down and its intervals?

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Answer 1 · 2015-09-07T01:52:25

See the explanation.

#f(x)=x^(4)-6x^(3)#

#f'(x) = 4x^3-18x^2#

#f''(x) = 12x^2-36x = 12x(x-3)#

#f''(x) = 0# at #0# and at #3#

On #(-oo,0)#, we have #f''(x) >0# so the graph of #f# is concave up.

On #(0,3)#, we have #f''(x) <0# so the graph of #f# is concave down.

On #(3,-oo)#, we have #f''(x) >0# so the graph of #f# is concave up.

Inflection points are #(0,0)# and #(3, -81)#