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

##### 1 Answer

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.

#### Explanation:

So, lets first take the derivative of

**Critical values**

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]}