Question

Question: 60-64 Find the limits as \(x \to \infty \) and as \(x \to - \infty \). Use this information together with intercepts, to give a rough sketch of the graph as in Example 12.

60. \(y = {\bf{2}}{x^{\bf{3}}} - {x^{\bf{4}}}\)

Short answer

The graph is shown below:

Step-by-step solution

Simplfy the function f

Simplify the function \(y = 2{x^3} - {x^4}\).

\(\begin{array}{c}y = 2{x^3} - {x^4}\\ = {x^3}\left( {2 - x} \right)\end{array}\)

The curve of f is passing through the origin.

The curve is intersecting the x axis at \(x = 0\) and \(x = 2\).

As \(x \to \infty \), \(y \to - \infty \) and as \(x \to - \infty \), \(y \to - \infty \).

Sketch the graph of the function

The figure below represent the curve of \(y = {x^3} - {x^4}\).

Thus, the sketch is obtained.