18) Find all numbers c that satisfy the conclusion of the Mean Value Theorem for the following function and interval. Enter the values in increasing order.

f(x)=2x6x+12,[1,4]

what does "2x6x" mean

0.45, 0.09, 0.48, 0.032, 0.5

To find the values of c that satisfy the conclusion of the Mean Value Theorem for the function f(x) = 2x^6 + 6x + 12 on the interval [1, 4], we first need to check if the function is continuous on this interval and differentiable on the open interval (1, 4).

The function f(x) = 2x^6 + 6x + 12 is a polynomial function, which is continuous and differentiable for all real values of x. Therefore, it satisfies the first condition.

To check if the function is differentiable on the open interval (1, 4), we need to find its derivative, f'(x).

f'(x) = d/dx (2x^6 + 6x + 12)
= 12x^5 + 6

Since f'(x) is also a polynomial, it is differentiable for all real values of x. Hence, the second condition is also satisfied.

Now, to find the values of c, we need to find the average rate of change of the function on the interval [1, 4].

The average rate of change, given by the Mean Value Theorem, is equal to the derivative of the function evaluated at some point c in the open interval (1, 4), which is:

f'(c) = (f(4) - f(1))/(4 - 1)
= (2(4^6) + 6(4) + 12 - 2(1^6) - 6(1) - 12)/(3)

Simplifying this expression:

f'(c) = (32784 + 24 + 12 - 12)/3
= 32808/3

So, the values of c that satisfy the conclusion of the Mean Value Theorem are the solutions to the equation f'(c) = 32808/3.

To find these values, we can set the derivative equal to 32808/3 and solve for c:

12c^5 + 6 = 32808/3

Multiplying both sides by 3 to get rid of the fraction:

36c^5 + 18 = 32808

Subtracting 18 from both sides:

36c^5 = 32808 - 18
= 32790

Dividing both sides by 36:

c^5 = 32790/36
= 910

Now, we need to find the fifth root of 910 to find the values of c. Since the problem asks for them in increasing order, we start with the smallest positive fifth root of 910.

Using a calculator, we find:

c ≈ 3.841

So, the values of c that satisfy the conclusion of the Mean Value Theorem for the function f(x) = 2x^6 + 6x + 12 on the interval [1, 4] are c ≈ 3.841.