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. \(y=x^{4}-4 x^{3}+16 x-16\)

Short answer

The graph of the function has critical points and points of inflection at \(x = 0, x = 2\). The function decreases until \(x = 0\), increases until \(x = 2\), and then decreases. Concavity changes at \(x = 0\) and \(x = 2\).

Step-by-step solution

Determine the Critical Points

Critical points are defined as points where the derivative is zero or undefined. The first step is to take the derivative of the function \(f'(x) = 4x^{3} - 12x^{2} + 16\). Then, we can set the derivative equal to zero and solve for x, which gives us the critical points \(x = 0, x = 2\). These points are where the function might have local minima, maxima or points of inflection.

Determine the Inflection Points

Points of inflection are points where the function changes concavity. To find these points, we need to find the second derivative of the function \(f''(x) = 12x^{2} - 24x\). Setting this second derivative equal to zero gives the potential points of inflection \(x = 0, x = 2\). These points should be tested in the second derivative to see where the function changes concavity.

Testing Intervals

To find out where the function is increasing or decreasing and where it is concave up or down, split up the number line into intervals using the critical points and inflection points. These intervals can then be tested in the first and second derivative. This will provide information about the behavior of the function on these intervals.

Sketch the Graph

Now use all the gathered information to sketch the graph. Plot the critical points and the inflection points. Use the information from the intervals to draw the curve of the function, paying attention to where the function increases or decreases and where it is concave up or down. Mark the points of inflection and relative minima or maxima on the graph.