# How do you find the critical points for #f(x)= (2x^2+5x+5)/(x+1)#?

The derivative is either 0 or undefined:

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To find the critical points of ( f(x) = \frac{2x^2 + 5x + 5}{x + 1} ), you need to follow these steps:

- Find the derivative of the function using the quotient rule.
- Set the derivative equal to zero and solve for ( x ).
- Check the second derivative test to confirm whether the critical points are maximum, minimum, or inflection points.

First, find the derivative:

[ f'(x) = \frac{(2x+5)(x+1) - (2x^2+5x+5)}{(x+1)^2} ]

[ f'(x) = \frac{2x^2 + 2x + 5x + 5 - 2x^2 - 5x - 5}{(x+1)^2} ]

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

Set ( f'(x) ) equal to zero:

[ -3 = 0 ]

Since this equation has no solutions, there are no critical points.

Therefore, the function ( f(x) = \frac{2x^2 + 5x + 5}{x + 1} ) has no critical points.

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