# How do you determine the intervals where #f(x)=x^4-x^2# is concave up or down?

Concave up:

Concave down:

Having two zeroes means we have 3 intervals to examine:

graph{x^4-x^2 [-1.923, 1.923, -0.963, 0.959]}

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To determine the intervals where ( f(x) = x^4 - x^2 ) is concave up or down, we need to find the second derivative of ( f(x) ) and then analyze its sign.

First derivative: ( f'(x) = 4x^3 - 2x )

Second derivative: ( f''(x) = 12x^2 - 2 )

To find the intervals where the function is concave up or down, set ( f''(x) = 0 ) and solve for ( x ). Then, test the intervals determined by these critical points with values of ( f''(x) ) in each interval to determine concavity.

Solve ( 12x^2 - 2 = 0 ) to find critical points.

( 12x^2 - 2 = 0 ) ( 12x^2 = 2 ) ( x^2 = \frac{1}{6} ) ( x = \pm \sqrt{\frac{1}{6}} )

Now, test the intervals created by these critical points with values of ( f''(x) ):

- Choose a test point in each interval and plug it into ( f''(x) ).
- Determine the sign of ( f''(x) ) in each interval.

If ( f''(x) > 0 ), the function is concave up in that interval. If ( f''(x) < 0 ), the function is concave down in that interval.

Finally, state the intervals where the function is concave up or down.

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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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