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

Answer 1

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

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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Answer 2

To determine if ( f(x) = \frac{-x^3 + x^2 + 2x - 11}{x - 1} ) is increasing or decreasing at ( x = 2 ), we can evaluate the derivative of the function at that point.

First, let's find the derivative of ( f(x) ): [ f'(x) = \frac{d}{dx}\left(\frac{-x^3 + x^2 + 2x - 11}{x - 1}\right) ]

Using the quotient rule, we get: [ f'(x) = \frac{(x - 1)(-3x^2 + 2x + 2) - (-x^3 + x^2 + 2x - 11)(1)}{(x - 1)^2} ]

Now, we evaluate ( f'(2) ): [ f'(2) = \frac{(2 - 1)(-3(2)^2 + 2(2) + 2) - (-2^3 + 2^2 + 2(2) - 11)(1)}{(2 - 1)^2} ] [ f'(2) = \frac{(1)(-12 + 4 + 2) - (-8 + 4 + 4 - 11)}{(1)^2} ] [ f'(2) = \frac{-6 + 3}{1} ] [ f'(2) = -3 ]

Since ( f'(2) < 0 ), ( f(x) ) is decreasing at ( x = 2 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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