# For what values of x is #f(x)= (x-x^3)/(2-x^3)# concave or convex?

f(x) is concave down in (0,

&

concave up in (0,

Like wise for all x<0, f''(x) would be positive and hence concave up.

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To determine the concavity or convexity of ( f(x) = \frac{x - x^3}{2 - x^3} ), we need to find its second derivative and then analyze its sign.

First, find the first derivative of ( f(x) ) and the second derivative by differentiating again. Then, set the second derivative equal to zero to find the inflection points.

The first derivative of ( f(x) ) is: [ f'(x) = \frac{(2 - x^3)(1 - 3x^2) - (x - x^3)(-3x^2)}{(2 - x^3)^2} ]

The second derivative of ( f(x) ) is: [ f''(x) = \frac{6x^4 - 4x^2 - 6x^2 + 2}{(2 - x^3)^3} ]

Now, set ( f''(x) = 0 ) and solve for ( x ) to find the inflection points.

[ 6x^4 - 10x^2 + 2 = 0 ]

This is a quadratic equation in terms of ( x^2 ). Solve it to find ( x^2 ) and then find the corresponding values of ( x ).

Once you have the values of ( x ), analyze the sign of ( f''(x) ) in the intervals determined by the critical points and the boundary points (where the function is defined). If ( f''(x) > 0 ), the function is convex in that interval. If ( f''(x) < 0 ), the function is concave in that interval.

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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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- How do you determine the intervals where the graph of the given function is concave up and concave down #f(x)= sinx-cosx# for #0<=x<=2pi#?
- What are the inflection points for #f(x) = -x^4-9x^3+2x+4#?

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