Is #f(x)=(3x^3-5x^2+13x-11)/(x-3)# increasing or decreasing at #x=2#?

1 Answer
Roella W.
Jan 11, 2016

Increasing

Explanation:

Use the quotient rule of differentiation to find the gradient. Then if the gradient is negative the function is decreasing, and if it is positive the function is increasing.
If #f(x) = g(x)/h(x)#
#f'(x) =(g(x)h'(x) - g'(x)h(x))/(h^2(x))#

#f'(x) = ((3x^3 - 5x^2 +13x -11)(1) - (9x^2 -10x +13)(x-3))/(x-3)^2#
#=((3x^3 - 5x^2 +13x -11) - (9x^3-10x^2+13x-27x^2+30x-39))/(x-3)^2#
#=(-6x^3 + 32x^2 -30x+28)/(x-3)^2#
#=(-2(3x^3 -16x^2+15x-14))/(x-3)^2#
Substituting in #x=2# gives #(-2(24-64+30-14))/1#
#=-2(54 - 78) = -2*(-24) = 48#

The gradient (slope) of #48# is positive and the function is therefore increasing.

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