We will answer this question assuming that the exact values for y in the options are already given in the question since they look as valid values for y.
To answer this question, we need to remember that the local maximum of a function is the highest point in the range of the function in a particular section of the given graph. Likewise, the local minimum is the lowest point in the range of the function in a particular section of the graph.
As a first step to finding such maximum or minimum points, we can find the first derivative of the function and equates it to zero. The obtained value for x can be substituted into the original function to find such points.
However, we have the graph of the function here. We can see that the maximum and minimum values are the following:
Therefore, we have:
Interval [-3, 0]
If we see the graph, on the closed interval [ -3, 0] the value of the minimum is near -16. Since there is an option of y = -16.18, this is the local minimum for the closed interval [-3, 0].
Interval [0, 3]
Local maximum:
On this interval, we can see a local maximum near y = 4. From the options, we have that one of the given values is y = 3.75, and this looks like a good approximation for the local maximum of the function on this interval.
Local minimum:
We can observe that the local minimum corresponds to a value of y at the middle path between -2 and -4, and this must be y = -3 (if we can get it from the options).
In summary, we have that:
• On the interval [-3, 0] ---> local minimum is f(x) = -16.18.
,
• On the interval [0, 3] ---> local maximum is f(x) = 3.75.
,
• On the interval [0, 3] ---> local minimum is f(x) = -3.