No problem! Let's break down the problem step by step and work through it together.
(a) To find the critical number of the function f(x), we need to find the value of x where the derivative of f(x) equals zero.
1. Start by taking the derivative of f(x) with respect to x:
f'(x) = K(1 + c^2*x^3) * (3*c^2*x^2) * (1 + x)^3 - K(1 + c^2*x^3) * 3*(1 + x)^2
2. Set f'(x) equal to zero and solve for x:
0 = K(1 + c^2*x^3) * (3*c^2*x^2) * (1 + x)^3 - K(1 + c^2*x^3) * 3*(1 + x)^2
Now, you can factor out the common terms and simplify the equation. You should end up with an equation in terms of x. Solve this equation to find the critical number x.
Once you have the critical number x, you can use the second derivative test to classify it as a relative maximum or a relative minimum. The second derivative test involves evaluating the second derivative of f(x) at the critical number and checking its sign. If the second derivative is positive, it's a relative minimum. If the second derivative is negative, it's a relative maximum. If the second derivative is zero, then the test is inconclusive.
(b) Now let's move on to the next part of the problem - finding the largest and smallest values of f(x) for the given values of c, K, and the domain.
Plug in the values of c = 1, K = (2π/3), and the domain (sqrt(2) − 1) <= x <= 1 into the formula for f(x). Calculate f(x) for the lower bound of the domain and the upper bound of the domain. Compare the results to find the largest and smallest values of f(x).
(c) Repeat the same process as part (b) but now use the values c = sqrt(2), K = sqrt(3π/16), and the domain 0 <= x <= 1 for β-cristobalite.
(d) The last part of the problem asks us to find the limit of f(x) as x approaches infinity (lim f(x) as x→∞).
To evaluate this limit, substitute infinity into the formula for f(x) and see what it approaches. You may have to consider the behavior of the terms in the formula as x gets larger.
I hope this breakdown helps! If you have any more specific questions or need further assistance, feel free to ask.