Question

Use a graphing utility to graph the function. Choose a window that allows all relative extrema and points of inflection to be identified on the graph. \(y=3 x^{2 / 3}-2 x\)

Short answer

The graph of the function \(y=3x^{2/3}-2x\) has a relative minimum at \(x=1\) and an inflection point at \(x=1\).

Step-by-step solution

Find the derivative of the function

The first derivative of the function gives us the slope at a given point. In this case, differentiated function is \(y' = (2/3)*3x^{-1/3}-2 = 2x^{-1/3}-2\).

Find the critical points

Critical points occur where the derivative is zero or undefined. To find the critical points, solve the equation \(2x^{-1/3}-2=0\). At \(x=1\), the derivative is equal to 0. There’s no point at which the derivative does not exist. Hence, \(x=1\) is the critical point of the function.

Finding the Extrema

To know if this critical point is a maximum or minimum, we need to examine the sign of the derivative. In the intervals \(-\infty , 1\) and \(1, +\infty\), the first derivative changes its sign. Hence, we can conclude that \(x=1\) is a point of local extremum.

Find the second derivative of the function

The second derivative helps us to identify whether the extrema is a maximum or minimum and they also give the points of inflection. The second derivative of the function is \(y''=-2/3*x^{-4/3}\).

Use the second derivative to determine nature of extrema and find inflection points

For \(x<1\), \(y''>0\) which implies the function is concave up and for \(x>1\), \(y''<0\) which implies the function is concave down. Thus, \(x=1\) is an inflection point. And based on the second derivative test, for \(x=1\), \(y''\) is undefined. So, \(x=1\) is a point of minimum.

Graph the function using a graphing utility

Using a graphing utility, plot the function \(y=3x^{2/3}-2x\). Ensure the window allows all relative extrema and points of inflection to be identified.