Sketch the curve

y = x^3 - 12x^2 + 36x

I already did the question properly it is concave down at (0,4) and concave up at (4,infinity)

I found the abs max to be (2,32) and abs min to be (6,0)

What I wanted to know is when labeling the graph should I label the max/min as abs/local max or just abs max? I am having trouble figuring out if it is just an abs or a local.

Go on: wolframalpha dot com

When page be open in rectangle type:

x^3 - 12x^2 + 36x

and click option =

After few seconds you will see everything about that function.

To determine whether a maximum or minimum point on a graph is absolute or local, you need to analyze the behavior of the function around that point. Here's how you can do that for this particular curve:

1. Calculate the derivative of the function: y' = 3x^2 - 24x + 36.

2. Set the derivative equal to zero to find the critical points: 3x^2 - 24x + 36 = 0. You can factor this equation as (x - 2)(x - 6) = 0.

3. Solve for x to find the x-coordinates of the critical points: x - 2 = 0 or x - 6 = 0. Hence, x = 2 or x = 6.

4. Substitute the critical points into the original function to find the corresponding y-coordinates: For x = 2, y = (2)^3 - 12(2)^2 + 36(2) = 8 - 48 + 72 = 32. For x = 6, y = (6)^3 - 12(6)^2 + 36(6) = 216 - 432 + 216 = 0.

Now let's analyze the behavior of the curve:

- At x = 2, the y-coordinate is 32. To determine whether this is an absolute maximum or minimum, we need to consider the values of y in the vicinity of x = 2. If you examine the graph and the behavior of the curve, you can see that there are no other points with y-values higher than 32 nearby. Hence, the point (2, 32) is an absolute maximum.

- At x = 6, the y-coordinate is 0. If you observe the behavior of the curve near x = 6, you'll notice that there are points with y-values both higher and lower than 0 nearby. Therefore, the point (6, 0) is a local minimum but not an absolute minimum.

To label the graph accurately, you can label the point (2, 32) as an absolute maximum and (6, 0) as a local minimum.