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A function f:R HR is said to be strictly increasing if f(x1) < f(12) whenever I]...
7. Consider the function f:R + R defined by f(x) = x < 0, 3 > 0. e-1/x2, Prove that f is differentiable of all orders and that f(n)(0) = 0 for all n e N. Conclude that f does not have a convergent power series expansion En Anx" for x near the origin. [We will see later in this class that this is impossible for holomorphic functions, namely being (complex) differentiable implies that there is always a convergent power...
PROVE: 4. If f : R → R is a strictly increasing function, f(0) = 0, a > 0 and b > 0, then
Prove this proposition please. 4.2.19) Proposition. Whenever a <b, there is a smooth function f satisfying f(2)= { € (0,1), a<:<b, VIVA *> 6. For obvious reasons, such a function is called a bump function. o M Figure 61. A bump function
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.
Now assume that f(0) = 0 and f'(0) = 0. Prove that if f is twice differentiable and If"(x) < 1 for all x E R then 22 Vx > 0, f(x) < 2
8. Consider ar? +5 I <3 f(0) = 12.c +b > 3 Determine a and b such that f is continuous and differentiable at x = 3. )={
It is known that f :(0,2) + R is a differentiable function such that \f'(x) < 5 for all x € (0,2). Now let bn := f(2 – †) for all n € N. Prove that this is a Cauchy sequence.
It is known that f :(0,2) + R is a differentiable function such that \f'(x) < 5 for all x € (0,2). Now let bn := f(2 – †) for all n € N. Prove that this is a Cauchy sequence.
Ja i3D1 3. A function is called increasing on an interval I if f(x) < f(y) for all in I. Suppose f is increasing on [a, b). Partition [a, b] into n equal pieces, each of width A = (b – a)/n. Find simple upper and lower bounds for Ja Ef(Ti)A. (Hint: In each [pi-1 – pi], f(pi-1) < f(x) < f(p;); also note that i=1 L(pi) – f(pi-1) = f(b) – f(a).) alsvmin ba i=1 taoo) ai eil diod...
*5. A function f defined on an interval I = {x: a <x<b> is called increasing f(x) > f(x2) whenever xi > X2 where x1, x2 €1. Suppose that has the inter- mediate-value property: that is, for each number c between f(a) and f(b) there is an x, el such that f(x) = c. Show that a functionſ which is increasing and has the intermediate-value property must be continuous. This is from my real analysis textbook, We are establishing the...