1. Use the Mean Value Theorem to prove the following inequalities(a) sin x < x for x > 0(b) tan x > x for 0 < x < π22. Use the MVT to prove the inequality | sin a − sin b| ≤ |a − b| for all aand b.3. Show that √1 + x < 1 + 12 xif x > 0.

(a) According to the Mean Value Theorem, there exists a number c in (0, x) such that cos(c) = (sin(x) - sin(0)) / (x - 0). Since 0 < cos(c) <= 1 for 0 < c < x, we have sin(x) / x <= 1. Since sin(x) is positive for x > 0, we have sin(x) < x.

(b) Similarly, there exists a number c in (0, x) such that sec^2(c) = (tan(x) - tan(0)) / (x - 0). Since sec^2(c) > 1 for 0 < c < π/2, we have tan(x) / x > 1. Since tan(x) is positive for 0 < x < π/2, we have tan(x) > x.

According to the Mean Value Theorem, there exists a number c between a and b such that cos(c) = (sin(a) - sin(b)) / (a - b). Since -1 <= cos(c) <= 1, we have |sin(a) - sin(b)| <= |a - b|.

Let f(x) = √1 + x - 1 - x/2. Then f'(x) = 1/(2√1 + x) - 1/2. Since f'(x) < 0 for x > 0, f(x) is decreasing for x > 0. Since f(0) = 0, we have f(x) < 0 for x > 0, which means √1 + x < 1 + x/2 for x > 0.