Question

Find the absolute maximum and absolute minimum for the function on a given interval

Answer 1

To answer this question, we need to find the first derivative of this function and evaluate it in the given interval.

We have the function f(x) = 3x² - 24, and we need to find the absolute extrema on the interval [0, 7].

Why do we need to find the first derivative of that function?

The first derivative gives us the slope of a tangent at a point. The maximum or minimum points of a function have a derivative equal to zero since the slope of a line parallel to the x-axis is zero. This also tells us that the point is a maximum or minimum point for the function.

Derivative of the function

To find the first derivative, we can proceed as follows:

`(d(3x^2-24x))/(dx)=3(d(x^2))/(dx)-24(d(x))/(dx)=3(2x)-24(1)`

We applied the rules for the derivative of a square = 2x, the derivative of line x is 1, and we need to multiply them by the constants.

Then, the first derivative is

`(d(3x^2-24x))/(dx)=6x-24`

If we equate this derivative to zero (for the given reasons above):

`6x-24=0\Rightarrow6x=24\Rightarrow x=(24)/(6)\Rightarrow x=4`

Then, we have an extremum at x = 4. We need to evaluate the function for x = 4, as follows:

`f(4)=3(4)^2-24(4)=3\cdot16-96=48-96\Rightarrow f(4)=-48`

We can see this using the following graph for the function:

Therefore, we can say that we have an absolute minimum for the function on the interval [0, 7] is (4, -48). The number x = 4 is the only critical point found, and it is on this interval.

To find the absolute maximum on this interval, we can use the values for x in the interval. We need to use the endpoints to find it:

x y = f(x)

0 0

4 -48

7 -21

For the given interval, for the closed given interval, we have an absolute maximum at point (0, 0), as we can see in the function. For the entire function, there is not an absolute maximum. We only found an absolute minimum for this polynomial function.

In summary, if we are strict about the concepts, we can say that we have an absolute minimum for the function at (4, -48), and it is in the interval [0, 7], and we have a maximum on this interval at (0, 0).

Therefore, we can say that absolute maximum and minimum on the closed interval [0, 7] are:

• Absolute Maximum: (0, 0)

,

• Absolute Minimum: (4, -48)

[If we are strict about the concepts, the point (0, 0) could be treated as a local maximum.]