Is #f(x)=x^3+2x^2-4x-12# concave or convex at #x=3#?

1 Answer
Shwetank Mauria
Mar 17, 2016

At #f(3)# the function is convex.

Explanation:

A concave function is a function in which no line segment joining two points on its graph lies above the graph at any point.

A convex function, on the other hand, is a function in which no line segment joining two points on the graph lies below the graph at any point.

It means that, if #f(x)# is more than the average of #f(x+-lambda)# than the function is concave and if #f(x)# is less than the average of #f(x+-lambda)# than the function is convex.

Hence to find the convexity or concavity of #f(x)=x^3+2x^2-4x-12# at #x=3#, let us evaluate #f(x)# at #x=2.5, 3 and 3.5#.

#f(2.5)=(5/2)^3+2(5/2)^2-4(5/2)-12=125/8+50/4-10+12=(125+100-80-96)/8=49/8#

#f(3)=(3)^3+2(3)^2-4(3)-12=27+18-12-12=21#

#f(3.5)=(7/2)^3+2(7/2)^2-4(7/2)-12=343/8+98/4-14-12=(343+196-112-96)/8=331/8#

The average of #f(2.5)# and #f(3.5)# is #(49/8+331/8)/2=380/(2xx8)=95/4=23 3/4#

As, this is more than #f(3)#, at #f(3)# the function is convex.

graph{x^3+2x^2-4x-12 [-5, 5, -20, 30]}

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