# How do you use the first and second derivatives to sketch #h(x)=x³-3x+1#?

First derivative will give us critical points:

Points that represent potential local maximum/minimum values or aymptotes

Plugging in values on either side of these critical points will show how the curve is acting (ie. positive/negative slopes)

Second derivative will give us inflection points:

Points in which the curve changes concavity

...As well as tell us if the curve is concave up/down.

So, lets first take the derivative of

Check the intervals around the critical values:

For

For

For

This should make sense since we started with a cubic function.

These changes in slope indicate local extrema.

Local maximum at

Local minimum at

Now, lets take the second derivative:

Check the intervals around this value.

For

For

Thus, there is an inflection point at x=0.

Now that we have all that information, we should gather up some basic information to help us plot.

Let's find some intercepts by plugging in 0 for x into the original function f(x).

y-intercept

Plug in our extrema values and plot those.

Now, we know how the graph acts based on those intervals:

Increasing from

Decreasing from

Now concavity; since we know that the function is differentiable along the

Concave down from

Concave up from

graph{x^3-3x+1 [-5.304, 5.796, -2, 3.546]}

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To sketch the function ( h(x) = x^3 - 3x + 1 ) using its first and second derivatives:

- Find the first derivative ( h'(x) ) by differentiating ( h(x) ) with respect to ( x ).
- Find the critical points by setting ( h'(x) ) equal to zero and solving for ( x ). These points represent potential local extrema or inflection points.
- Determine the intervals of increase and decrease of ( h(x) ) by examining the sign of ( h'(x) ) in between the critical points.
- Find the second derivative ( h''(x) ) by differentiating ( h'(x) ) with respect to ( x ).
- Determine the concavity of ( h(x) ) by examining the sign of ( h''(x) ) between critical points.
- Locate any points of inflection where ( h''(x) = 0 ) or is undefined.
- Sketch the graph, incorporating the information about critical points, intervals of increase/decrease, and concavity.

By following these steps, you can construct a rough sketch of the function ( h(x) ).

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