Is #f(x)=x^4-2x^3-9x-14# concave or convex at #x=-1#?

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Answer 1 · 2015-12-25T09:15:26

At #x=-1# the curve is Convex or Concave Up Explanation given below.

#f(x) = x^4-2x^3-9x-14#

Step 1: Find the derivative

#f'(x) = 4x^3-6x^2-9#

Step 2: Differentiate again with respect to x

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

Step 3: Substitute #x=-1# in #f^2(x)# and check for sign

#f^2(-1)=12(-1)^2-12(-1)#
#f^2(-1)=12+12#
#f^2(-1)=24#

If #f^2(x) >0# then the curve is convex.
If #f^2(x) <0#, then the curve is *concave *

We can see at #x=-1# the second derivative is greater than zero, hence, the curve is convex.

For further information, you can refer
Note: Concave up is same as convex and concave down is concave