Question

type oo for oo and -oo for -oo. also type u for u.

what is the global minimum of f? where does it occur?
where is f undefined?
the global minimum is -1.2 which
at x = dne occurs at x = -2.2

what are the critical values of f?
what is the global maximum of f? where does it
at x = -2,-1,0,1,2 occur?
the global maximum is dne which

where does f have local minimums?
occurs at x = -0
at x = -2,2

where does f have local maximums?
where is f>0?
at x = -1,1 ∀x in the interval (-2,-0.2) u (1,2)

where is f<0?
∀x in the interval (-3,-2) u (-0.2,1) u (2,3)

Explanation:

Step1: Analyze the graph for critical points

Critical points occur where \(f^{\prime}(x)=0\) or \(f^{\prime}(x)\) is undefined. From the graph, \(f^{\prime}(x)\) is defined for all \(x\) (so \(x = \text{DNE}\) for \(f^{\prime}\) undefined). The critical values (where \(f^{\prime}(x) = 0\)) are at \(x=-2,-1,0,1,2\).

Step2: Identify local minima and maxima

Local minima occur where the function changes from decreasing to increasing. From the graph, this is at \(x = - 2\) and \(x=2\). Local maxima occur where the function changes from increasing to decreasing, which is at \(x=-1\) and \(x = 1\).

Step3: Determine global minimum and maximum

The global minimum is the lowest value of the function. From the graph, the global minimum value is \(-1.2\) (assuming the lowest \(y\) - value) which occurs at \(x=-2\) and \(x = 2\). Since the function tends to \(+\infty\) as \(x\to\pm\infty\) (from the graph's end - behavior, as the graph goes up on both ends), there is no global maximum (\(\text{DNE}\)).

Step4: Find intervals where \(f^{\prime}(x)>0\) and \(f^{\prime}(x)<0\)

Using the first - derivative test (relationship between the slope of the function and its derivative):

• For \(f^{\prime}(x)>0\) (function is increasing), we look at the intervals where the function is rising. From the graph, \(f(x)\) is increasing on \((-2,-1)\cup(1,2)\) (assuming the correct intervals based on the graph's slope direction).
• For \(f^{\prime}(x)<0\) (function is decreasing), we look at the intervals where the function is falling. From the graph, \(f(x)\) is decreasing on \((-3,-2)\cup(-1,1)\cup(2,3)\) (assuming the correct intervals based on the graph's slope direction).

Answer:

• Where is \(f^{\prime}\) undefined? \(x=\text{DNE}\)
• What are the critical values of \(f\)? \(x=-2,-1,0,1,2\)
• Where does \(f\) have local minima? \(x=-2,2\)
• Where does \(f\) have local maxima? \(x=-1,1\)
• What is the global minimum of \(f\)? The global minimum is \(-1.2\) which occurs at \(x=-2,2\)
• What is the global maximum of \(f\)? The global maximum is \(\text{DNE}\)
• Where is \(f^{\prime}>0\)? \(\forall x\) in the interval \((-2,-1)\cup(1,2)\)
• Where is \(f^{\prime}<0\)? \(\forall x\) in the interval \((-3,-2)\cup(-1,1)\cup(2,3)\)