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Is #f(x)=-x^5+3x^4-9x^3-2x^2-6x# concave or convex at #x=8#?

1 Answer
Apr 1, 2017

Answer:

#f(x)# is concave at #x=8#. We know this by looking at the second derivative, which tells us about the concavity/shape of the graph.

Explanation:

When looking for the concavity of a function, it's best to find the second derivative, #f''(x)#, of the function, #f(x)#.

When #f''(x)<0#, the #f(x)# is concave
When #f''(x)>0#, the #f(x)# is convex

The first derivative of this function is:
#f'(x)=-5x^4+12x^3-27x^2-4x-6#

The second derivative is:
#f''(x)=-20x^3+36x^2-54x-4#

Plug in #x=8# to get:
#f''(8)=-20(8)^3+36(8)^2-54(8)-4#
#f''(8)=-8372#

Since #f''(8)<0#, the #f(x)# is concave at #x=8#.