Reference no: EM131084850
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 limx→∞ 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 limx→∞ 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).
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