For what values of x is #f(x)=3x^3-7x^2-5x+9# concave or convex?

1 Answer
mason m
Jun 5, 2016

#f# is concave (concave down) on #(-oo,7/9)# and is convex (concave up) on #(7/9,oo)#.

Explanation:

The convexity and concavity of the function #f# can be determined by looking at the sign of the second derivative:

  • If #f''>0#, then #f# is convex.
  • If #f''<0#, then #f# is concave.

To find the function's second derivative, use the power rule.

#f(x)=3x^3-7x^2-5x+9#

#f'(x)=9x^2-14x-5#

#f''(x)=18x-14#

So, the convexity and concavity are determined by the sign of #f''(x)=18x-14#.

The second derivative equals #0# when #18x-14=0#, which is at #x=7/9#.

When #x>7/9#, #f''(x)>0#, so #f(x)# is convex on #(7/9,oo)#.

When #x<7/9#, #f''(x)<0#, so #f(x)# is concave on #(-oo,7/9)#.

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