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

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Answer 1 · 2016-01-23T16:46:53

#f(x)=(x-3)(x+2)(3x-2)#
#implies f(x)=(x^2-x-6)(3x-2)#
#implies f(x)=3x^3-5x^2-4x+12#

If #f(x)# is a function and #f''(x)# is the second derivative of the function then,

#(i) f(x)# is concave if #f(x)<0#
#(ii) f(x)# is convex if #f(x)>0#

Here #f(x)=3x^3-5x^2-4x+12# is a function.

Let #f'(x)# be the first derivative.
#implies f'(x)=9x^2-10x-4#

Let #f''(x)# be the second derivative.
#implies f''(x)=18x-10#

#f(x)# is concave if #f''(x)<0#
#implies 18x-10<0#
#implies 9x-5<0#
#implies x<5/9#

Hence, #f(x)# is concave for all values belonging to #(-oo,5/9)#

#f(x)# is convex if #f''(x)>0#.
#implies 18x-10>0#
#implies 9x-5>0#
#implies x>5/9#

Hence, #f(x)# is convex for all values belonging to #(5/9,oo)#