Question

2-50. using the graph at right, where is the function increasing/decreasing? where is the function concave up/down? does this function have any maxima or minima? explain how you know. homework help

Explanation:

Step1: Determine increasing - decreasing intervals

For a function \(y = f(x)\), if the slope of the tangent line is positive, the function is increasing; if negative, it is decreasing. Looking at the graph, as \(x\) increases from \(-\infty\) to \(0\), the \(y\) - values of the function are decreasing. As \(x\) increases from \(0\) to \(\infty\), the \(y\) - values are increasing. So the function is decreasing on the interval \((-\infty,0)\) and increasing on the interval \((0,\infty)\).

Step2: Determine concavity

The concavity of a function can be determined by the second - derivative (or by the shape of the graph). A function is concave up if it "holds water" and concave down if it "spills water". The graph of the given function is curved upwards, so it is concave up on the interval \((-\infty,\infty)\).

Step3: Determine maxima and minima

A local minimum occurs at a point where the function changes from decreasing to increasing. Since the function changes from decreasing to increasing at \(x = 0\), there is a local (and in this case, global) minimum at \(x=0\). The value of the minimum is \(y = 0\) (the \(y\) - value of the function at \(x = 0\)). There are no local or global maxima.

Answer:

• Increasing interval: \((0,\infty)\)
• Decreasing interval: \((-\infty,0)\)
• Concave - up interval: \((-\infty,\infty)\)
• Concave - down interval: None
• Maxima: None
• Minima: The function has a minimum at \(x = 0\) with \(y=0\)