# What are the values and types of the critical points, if any, of #f(x)=(x+3x^2) / (1-x^2)#?

There is a local maximum at

The function is

The derivative of a quotient is

Therefore,

The critical points are when

That is

The solutions to this quadratic equation are

and

Let's build a variation chart

Therefore,

graph{(x+3x^2)/(1-x^2) [-10, 10, -5, 5]}

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To find the critical points of ( f(x) = \frac{x + 3x^2}{1 - x^2} ), first find its derivative, ( f'(x) ), and then solve for ( x ) when ( f'(x) = 0 ) or ( f'(x) ) is undefined.

( f'(x) = \frac{d}{dx} \left( \frac{x + 3x^2}{1 - x^2} \right) )

( f'(x) = \frac{(1 - x^2)(6x) - (x + 3x^2)(-2x)}{(1 - x^2)^2} )

( f'(x) = \frac{6x - 6x^3 + 2x^2 - 6x^3}{(1 - x^2)^2} )

( f'(x) = \frac{-12x^3 + 2x^2 + 6x}{(1 - x^2)^2} )

Now, set ( f'(x) ) equal to zero and solve for ( x ) to find critical points:

( -12x^3 + 2x^2 + 6x = 0 )

( 2x(-6x^2 + x + 3) = 0 )

( 2x(2x - 3)(-3x - 1) = 0 )

( x = 0, \frac{3}{2}, -\frac{1}{3} )

To determine the type of critical points, you can use the first or second derivative test. Calculate ( f''(x) ) to apply the second derivative test:

( f''(x) = \frac{d}{dx} \left( \frac{-12x^3 + 2x^2 + 6x}{(1 - x^2)^2} \right) )

( f''(x) = \frac{(-12x^3 + 2x^2 + 6x)(2)(1 - x^2)^2 - (-12x^3 + 2x^2 + 6x)(-2)(1 - x^2)(-2x)}{(1 - x^2)^4} )

Evaluate ( f''(x) ) at the critical points ( x = 0, \frac{3}{2}, -\frac{1}{3} ) to determine the nature of each critical point.

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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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