Assignment 5 an square abcd with side lengths 4 is given

Question

Assignment 5-

1. Use the Mean Value Theorem to show that

1/9 < √66 - 8 < 1/8.

2. In this problem, we will use the Mean Value Theorem to prove that if f(x) is a differentiable function defined on (0, ∞) such that both f(x) and f'(x) are strictly increasing, then lim_x→∞ f(x) = ∞.

(a) First, show that there is some value c ∈ (1, 2) such that f'(c) > 0.

(b) Now, show that for any x > 2 we have f(x) > f(2) + (x - 2)f'(c).

(c) Conclude that lim_x→∞ f(x) = ∞.

3. Graph the function

y = x⁴/81 - 2x²/9,

and label and characterize all critical points.

4. An square ABCD with side lengths 4 is given. Five circles are placed inside this square. Four of them have the same radius and are internally tangent to the sides meeting at vertices A, B, C and D respectively. The fifth is internally tangent to the other four circles. Determine the maximum and minimum possible total area of all these circles.

5. Suppose f(x) is a differentiable function with f'(x) = (1 + x³)^-1/2. Let g(x) be the inverse of f(x). Assuming g(x) is twice differentiable, show that 2 · g''(x) = 3 · g²(x).