Question

Use the mean value theorem to show the following: (a) 1+{h/[2(1+h)]}⩽ √1+h ⩽1+h/2

Answer 1

To use the mean value theorem to show the inequality 1 + h/[2(1+h)] ≤ √(1+h) ≤ 1 + h/2, apply the theorem to the function f(x) = √x on the interval [1, 1+h].

Step-by-step explanation:

To use the mean value theorem to show the inequality 1 + h/[2(1+h)] ≤ √(1+h) ≤ 1 + h/2, we can apply the theorem to the function f(x) = √x on the interval [1, 1+h].

1. By the mean value theorem, there exists a c in (1, 1+h) such that f'(c) = [f(1+h) - f(1)]/(1+h - 1).
2. Simplifying, we have 1/√c = (√(1+h) - √1)/h.
3. Multiplying both sides by h and rearranging terms, we obtain 1/√c * h = (√(1+h) - 1)/h.
4. Since 1 < c < 1+h, we can say that 1/√(1+h) * h ≤ (√(1+h) - 1)/h ≤ 1/√1 * h.
5. Further simplifying, we get 1/[2(1+h)] ≤ (√(1+h) - 1)/h ≤ 1/2.
6. Multiplying all terms by h, we have 1 + h/[2(1+h)] ≤ √(1+h) ≤ 1 + h/2. Therefore, we have shown the desired inequality.