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

At

Let's compute (using the division derivatove rule):

Next, let's replace x=1:

By signing up, you agree to our Terms of Service and Privacy Policy

To determine if ( f(x) = \frac{-x^3 - 2x^2 - x + 2}{x^2 + x} ) is increasing or decreasing at ( x = 1 ), we can use the first derivative test. First, we find the derivative of ( f(x) ), and then we evaluate the sign of the derivative at ( x = 1 ). If the derivative is positive, the function is increasing at that point. If it's negative, the function is decreasing.

The derivative of ( f(x) ) is ( f'(x) = \frac{d}{dx} \left( \frac{-x^3 - 2x^2 - x + 2}{x^2 + x} \right) ).

To simplify this derivative, we can use the quotient rule:

[ f'(x) = \frac{(x^2+x)(-3x^2-4x-1) - (-x^3-2x^2-x+2)(2x+1)}{(x^2+x)^2} ].

After simplifying, we get:

[ f'(x) = \frac{-x^4 - 4x^3 - 3x^2 + 2x^3 + 8x^2 + 2x + 2x^3 + 4x^2 + x - 2}{(x^2+x)^2} ] [ f'(x) = \frac{-x^4 - 2x^3 + 9x^2 + 3x - 2}{(x^2+x)^2} ].

Now, we can evaluate ( f'(1) ) to determine if the function is increasing or decreasing at ( x = 1 ):

[ f'(1) = \frac{-1^4 - 2(1)^3 + 9(1)^2 + 3(1) - 2}{((1)^2 + 1)^2} ] [ f'(1) = \frac{-1 - 2 + 9 + 3 - 2}{(1 + 1)^2} ] [ f'(1) = \frac{7}{4} ].

Since ( f'(1) = \frac{7}{4} > 0 ), the function ( f(x) ) is increasing at ( x = 1 ).

By signing up, you agree to our Terms of Service and Privacy Policy

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.

- Is #f(x)=(1-e^(2x))/(2x-4)# increasing or decreasing at #x=-1#?
- What are the values and types of the critical points, if any, of #f(x,z) = x^4 + 15z^2 + 2xz^2 - 456z^2#?
- What are the critical points of #f(x) =x^(x^2)#?
- What are the critical values of #f(x)=xe^(2x+5)-x^3e^(-x^2-x)#?
- What are the absolute extrema of #f(x)=(x-2)(x-5)^3 + 12in[1,4]#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7