Prove this proposition please. 4.2.19) Proposition. Whenever a <b, there is a smooth function f satisfying...
A function f:R HR is said to be strictly increasing if f(x1) < f(12) whenever I] < 12. Prove: If a differentiable function f is strictly increasing, then f'(x) > 0. Then give counterexamples to show that the following statements are false, in general. (i) If a differentiable function f is strictly increasing, then f'(2) >0 for all 1. (ii) If f'(x) > 0 for all x, then f is strictly increasing -
(c) A sequence {2n} satisfying 0 < In < 1/n where E(-1)"In diverges.
Exercise 5. Prove that if f is a continuous and positive function on (0,1], there exists 8 >0 such that f(x) > 8 for any x € [0,1].
Prove that if |A| = |Band [B<|A|, then |A| = |B).
4. Let F be a continuously differentiable function, and let s be a fixed point of F (a) Prove if F,(s)| < 1, then there exists α > 0 such that fixed point iterations will o E [s - a, s+a]. converge tO s whenever x (b) Prove if IF'(s)| > 1, then given fixed point iterations xn satisfying rnメs for all n, xn will not converge to s.
3) Prove that there exists f : R → R non-negative and continuous such that f € L'OR : dm) ( i.e. SR \f|dm <00) and lim sup f(x) = ∞. 2-0
Please Prove the Following: Prove that if A is a finite set (i.e. it contains a finite number of ele ments), then IAI < INI, and if B s an infinite set, then INI-IBI
Please don't use schwarz pick lemma 5.17. Suppose f : D[0,1] → D[0,1] is holomorphic. Prove that for z1 <1, 1 |f'(2) 1 - 12
7. (5 marks) Consider a smooth function u(x, y) satisfying: Urx + Uyy + Uzy > 0 in 12. . Show that u atains its maximum on the month ago Show that u attains its maximum on the boundary an.
The directional derivative of the function f(x, y) = 2x In(y) in the direction v =< 0,1 > at the point (1,1) is equal to 2. Select one: O True False