4. (a) Assume a function h is differentiable at some point to. Is it true that...
3. Let f: R+R be a function. (a) Assume that f is Riemann integrable on [a, b] by some a < b in R. Does there always exist a differentiable function F:RR such that F' = f? Provide either a counterexample or a proof. (b) Assume that f is differentiable, f'(x) > 1 for every x ER, f(0) = 0. Show that f(x) > x for every x > 0. (c) Assume that f(x) = 2:13 + x. Show that...
Assume f : R" → R is twice continuously differentiable. Prove that the following are equivalent: (a) f(ex + (1-8)ì) < ef(x) + (1-8)/(x) for all x, x E Rn and 0 < θ < 1 (b) f(x)+ /f(x) . (x-x) -f(r) for all x,x E R" (c) f(x) > 0 for all x E R" Hint: Look at : RRdefine by gt) f(x + ty) where x, y E R. First show g is convex (as a function of...
Please prove by setting up the theorem below (Chain Rule) v:RR is continuously differentiable. Define the Suppose that the function function g : R2R by 8(s, t)(s2t, s) for (s, t in R2. Find ag/as(s, t) and ag/at(s, t) Theorem 15.34 The Chain Rule Let O be an open subset of R and suppose that the mapping F:OR is continuously differentiable. Suppose also thatU is an open subset of Rm and that the functiong:u-R is continuously differentiable. Finally, suppose that...
Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists an M R such that f(x) < f(xM) for al E R. Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists...
Suppose that g(x) is differentiable on (0,1) and g'(x)1 for any x (0,1). Show that if g(a)=a andg(b)=b for some a,b (0,1) then a=b.I dont know where to start with this.So by Rolle's Theorem: Let f : [a,b] becontinuous on [a,b] and differentiable on (a,b). If f(a)=f(b), then there exists a point c (a,b) where f'(c)=0. The proof forthis says that because f is continuous on a compact set, f attains a maximum and a minimum. If both the maximum...
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.
Let U be an open subset of R. Let f: U C Rn → Rm. (a) Prove that f is continuously differentiable if and only if for each a є U, for each E > 0, there exists δ > 0 such that for each x E U, if IIx-all < δ, then llDf(x)-Df(a) ll < ε. (b) Let m n. Prove that if f is continuously differentiable, a E U, and Df (a) is invertible, then there exists δ...
real analysis 1,2,3,4,8please 5.1.5a Thus iff: I→R is differentiable on n E N. is differentiable on / with g'(e) ()ain tained from Theorem 5.1.5(b) using mathematical induction, TOu the interal 1i then by the cho 174 Chapter s Differentiation ■ EXERCISES 5.1 the definition to find the derivative of each of the following functions. I. Use r+ 1 2. "Prove that for all integers n, O if n is negative). 3. "a. Prove that (cosx)--sinx. -- b. Find the derivative...
5. (This question shows an alternative definition for the term differentiable, one that is useful when you study advanced calculus (in R" instead of R).) Let f: R+R and a ER. Prove that f is differentiable at a if and only if there exists a function L:RR which is continuous at a and satisfies f(x) - f(a) = L(x)(x - a) for all x ER Suggestion: If f is differentiable at a, try to define the L function separately at...
Question 4. For S: B(ro, 0), assume that f: S R" is a function such that f(x) f(y)Plx - y f(0) c and for some pi1 a. Prove that for any x E S f(x) elpilx|< \cl + Pi*o b. Prove that there exists some rı > 0 such that c|< r1 implies f(x) e S for all x E S (Find a particular choice of ri that will work.) Question 4. For S: B(ro, 0), assume that f: S...