# How do you sketch the graph by determining all relative max and min, inflection points, finding intervals of increasing, decreasing and any asymptotes given #f(x)=x^4-4x^3#?

##### 1 Answer

See below.

#### Explanation:

**Asymptotes:** None. Polynomials don't have (linear) asymptotes.

**Intercepts**

**Analysis of #f'(x)#**

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There is a relative minimum at

There are no relative maxima.

Before we look at concavity, here is the straight line sketch:

**Analysis of #f''(x)#**

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Inflection points are points on the graph at which the concavity changed. Therefore, there are inflection points at

The inflection points are:

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Now that we have concavity, we can improve our sketch:

Here is Socratic's graph. (You can move it and zoom in/out. If you leave this answer and come back, the graph will reset to the original view.)

graph{x^4-4x^3 [-37.2, 35.86, -29.07, 7.48]}