# How many points of inflection does the function #f(x)=x^7-x^2# have?

Inflection points are where the function changes concavity.

The second derivative must equal zero when the function changes concavity.

But we must check points on either side to make sure that the concavity really does change.

So,

To make sure that the concavity actually changes, we pick a number on either side of

Let's use

The concavity changes, so

The brown square represents the inflection point.

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The function ( f(x) = x^7 - x^2 ) has 4 points of inflection.

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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.

- How do you find all points of inflection given #y=((x-3)/(x+1))^2#?
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- How do you sketch the graph #y=sinx+sin^2x# using the first and second derivatives from #0<=x<2pi#?
- For what values of x is #f(x)= x^4-3x^3-4x-7 # concave or convex?
- How do you determine whether the function #y=x^2 # is concave up or concave down and its intervals?

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