# How do you sketch the graph #y=x^3-4x^2# using the first and second derivatives?

- General Observations

A cubic function and the cubic coefficient is +ve, so it will have the classic cubic shape with a maximum to the left of a minimum,

- Roots

- Turning Points

- Nature of Turning Points

- The Graph

There is now information to plot the graph, here I will use the actual graph:

graph{x^3-4x^2 [-10, 10, -15, 10]}

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To sketch the graph of (y = x^3 - 4x^2) using the first and second derivatives:

- Find the critical points by setting the first derivative equal to zero and solving for (x).
- Use the second derivative test to determine the nature of the critical points (whether they are local maxima, local minima, or points of inflection).
- Identify any horizontal or vertical asymptotes, if applicable.
- Plot the critical points and any asymptotes on the graph.
- Determine the behavior of the function as (x) approaches positive and negative infinity.
- Sketch the graph based on the above information.

To summarize, you first find the critical points using the first derivative, then analyze the concavity using the second derivative, and finally incorporate additional information such as asymptotes and the behavior of the function at infinity to sketch the graph.

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