Find the minimum value of the function f(x) = -x^3 + 3ax on the interval 0 ≤ x ≤ 2. assume that a > 0. give the minimum value in terms of a.

for 0 ≤ a < , minimum is -8 + 6a

              for ≤ a ≤ 2, minimum is 0

Step-by-step explanation:

f(x) = -x³ + 3ax is continuous and differentiable on the interval [0, 2], so the minimum will occur at a critical point or an endpoint.

Step 1: Find the critical point(s) between [0, 2] by setting the derivative of the function equal to zero and solving for x.

f(x) = -x³ + 3ax

f'(x) = -3x² + 3a

0 = -3x² + 3a

3x² = 3a

x² = a

x = ±√a

x = √a   and   x = -√a

                          ↓

                      not valid because it is not in the interval [0, 2]

So, the only critical point is at x = √a

Step 2: Evaluate f(x) at the critical point (x = √a) and endpoints (x = 0 and x = 2)

f(x) = -x³ + 3ax

f(0) = -(0)³ + 3a(0)

     =  0 + 0

     = 0

f(√a) = -(√a)³ + 3a(√a)

        = -a√a + 3a√a

        = 2a√a

f(2) = -(2)³ + 3a(2)

     =  -8 + 6a

Step 3: Evaluate the values of "a" that will determine the minimum

Notice that f(a) > f(2) for all a-values so this can be disregarded for the minimum. So, we need to see when f(2) < f(0)

-8 + 6a < 0

       6a < 8

Thus, the minimum is -8 + 6a when "a" is between 0 (the endpoint) and .  The minimum is 0 when "a" is between and 2 (the endpoint).

Bonus: See the attached graph to see minimum for different values of "a".

as "a" approaches 0, y = -8 is the lowest value

when a = 1,  y = -2 is the lowest value

when a = ,  y = 0 is the lowest value

when a = 2,  y = 0 is the lowest value

a

step-by-step explanation:

the range of a function is the range of y-values for which it is defined, or exists. given that we are only looking at the y-values, we can already discount the first two answers as they deal with the domain of the function (range of x-values).

now, what we need to do is find the smallest and highest y-values for which the function is defined. if you'd like, you could take a ruler and start at the very bottom of the graph, find the lowest point, then move the ruler higher and higher until you reach the highest point. doing so either in your head or with an actual ruler, we can see that the lowest y-value is -5 and the highest y-value is -1.

the question now remains as to whether they are included in the range (fourth answer) or excluded (third answer). the key here is to recognise whether they are marked with an open or a closed circle. since both (-2, -5) and (2, -1) are marked with a closed circle, they are points that exist on the graph and are thus included in the range (note that if they were marked with an open circle like (4, -2) then that would signify that there is a break in the graph at that point).

thus, the fourth answer, (y | -5 ≤ y ≤ -1), is correct.

also note that usually it is good practice to scan your eye inbetween the highest and lowest point for any breaks in the graph - we do have one here at (4, -2), however -2 is still included in the range because there is another point that is included in the graph with the same y-value, ie. point (1, -2).

i hope that but if you require any further explanation don't hesitate to comment below.