Question

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=x^{3}-4 x^{2}+6\)

Short answer

The graph of the function \(y=x^{3}-4 x^{2}+6\) rises to the left and right. It has roots at \(x = 0\) and \(x = \frac{8}{3}\) (local extremes), as well as a possible point of inflection at \(x = \frac{4}{3}\). The actual sketch should show these points and the overall shape of a cubic function.

Step-by-step solution

Find the Root(s) or Zero(s)

For a polynomial function, the root or zero of the function can be found by setting \(y\) equal to zero and solving for \(x\). In this case: \[0= x^{3}-4 x^{2}+6\] Solving this cubic equation may be complex, hence we may also calculate roots accurately using numerical methods or graphically.

Compute the First Derivative

The first derivative represents the slope of the function and where the tangent line to the function is horizontal at points, there may be a local maximum or minimum. The derivative of \(y= x^{3}-4 x^{2}+6\) is: \[ y' = 3x^{2} - 8x\] Setting \(y'\) equal to zero and solving for \(x\) will give the critical points: \[0= 3x^{2} - 8x\] Hence, \(x = 0\) or \(x = \frac{8}{3}\).

Compute the Second Derivative

The second derivative, is used for identifying points of inflection and for determining whether each local maximum or minimum is indeed a maximum or minimum. The second derivative of the function is: \[y'' = 6x - 8\] Setting \(y'' = 0\), we get \(x = \frac{4}{3}\). \(\frac{4}{3}\) represents the possible point of inflection.

Sketch the Graph

Use the zeros, points of local maximums or minimums, and points of inflection from the previous steps to sketch the graph. Pay attention to the end behavior as \(x\) approaches positive or negative infinity. As \(x\) approaches negative and positive infinity, \(y = x^{3}-4 x^{2}+6\) will tend to negative and positive infinity too, respectively. Thus the function will rise to the right and left.