Is f(x)=(-x^3+x^2+2x-11)/(x-1)f(x)=−x3+x2+2x−11x−1 increasing or decreasing at x=2x=2?

1 Answer
benjidunn
Apr 24, 2016

Gradient is positive hence function is increasing at x=2x=2

Explanation:

Find f'(x) for increasing/decreasing.
Use quotient rule since the function is a fraction.

Let f=-x^3+x^2+2x-11 and g=x-1
f'=-3x^2+2x+2
g'=1
f'(x)=(f'g-g'f)/(g^2)
f'(x)=((-3x^2+2x+2)(x-1)-(1)(-x^3+x^2+2x-11))/(x-1)^2
f'(x)=(-2x^3+4x^2-2x+9)/(x-1)^2

f'(2)=(-2(2)^3+4(2)^2-2(2)+9)/((2)-1)^2
=(-16+16-4+9)/1
=5

Gradient is positive hence function is increasing at x=2

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