Final answer:
To find local maxima and minima, calculate the first derivative's zeros and use the second derivative test for confirmation. Plotting the function can visually confirm the local extremes.
Step-by-step explanation:
To find all the local maximum and minimum values of a function, you should first calculate the function's first derivative and find its zeros by solving for when the derivative equals zero. These points are the potential local maxima and minima. However, to confirm if these points are indeed maxima or minima, the second derivative test can be used. If the second derivative at a zero of the first derivative is positive, it is a local minimum. If it's negative, it is a local maximum. If the second derivative equals zero, the test is inconclusive, and further investigation is required.
For graphical analysis, plotting the function and identifying the peaks and troughs can provide visual confirmation of the local extremes. Intercepts, maximums, and minimums should be clearly labeled on the graph for better understanding.