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5. Let F : RP → R9 be a continuous function and let ECRP be closed....
3. Let f: RP-R (a) If f(x)-Ax + b, x E R A є Mq.p and b є R9, show that f is p. where differentia...
3. Let f: RP-R (a) If f(x)-Ax + b, x E R A є Mq.p and b є R9, show that f is p. where differentiable everywhere and calculate its total derivative (b) If f is differentiable everywhere and Df (x)A, for some A E Mp and all q.p x E Rp, show that there exists b E R, such that f(x) = Ax + b for all x E Rp
3. Let f: RP-R (a) If f(x)-Ax + b,...
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing funct...
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1) Show that the inverse function f -1 exists. (2) Prove that f is an open map (in the relative topology on I) (3) Prove that f1 is continuous
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1)...
23. Let be a function defined and continuous on the closed interval (a,b). If f has...
23. Let be a function defined and continuous on the closed interval (a,b). If f has a relative maximum at cand a<c<b, which of the following statements must be true? 1. f'(c) exists. II. If f'(c) exists, then f'(c)= 0. III. If f'(c) exists, then f"(c)<0. (A) II only (B) III only (C) I and II only (D) I and III only (E) II and III only
Question 4* (Similar to 18.1) Suppose f is a continuous function on a closed interval [a,...
Question 4* (Similar to 18.1) Suppose f is a continuous function on a closed interval [a, b]. In class, we proved that f attains its maximum on that interval, i.e. there exists Imar E la, so that f(Imar) > f(x) for all r E (a,b]. We didn't prove that f attains its minimum on the interval, but I claimed that the proof is similar. In fact, you can use the fact that f attains its maximum on any closed interval...
please explain in detail 4 -11 23 4 Graph of f Let f be a continuous function defined on the closed interval -1Sxs4. The graph of f, consisting of three line segments, is shown above. Let g be...
please explain in detail
4 -11 23 4 Graph of f Let f be a continuous function defined on the closed interval -1Sxs4. The graph of f, consisting of three line segments, is shown above. Let g be the function defined by g(x) = 5 +1.f(t) dt for-1 $154. (A) Find g(4). (B) On what intervals is gincreasing? Justify your answer. (C) On the closed interval 1 s xs 4, find the absolute minimum value of g and find the...
4. The function f is continuous on the closed interval (-2, 1). Some values of f...
4. The function f is continuous on the closed interval (-2, 1). Some values of f are shown in the table below. --2 f(x) -3 -1 0 1 7 k3 The equation f(x) = 3 must have at least two solutions in the interval [-1,1) if k = a. 1 b. C. 2 CONN NICO d. 5. If k(r) is a continuous function over the interval (-2, 4) such that k(-2) = 3 and k(4) = 1, then k(2) 0...
Let f be the function given by f () = on the closed interval [-7,7]. Of...
Let f be the function given by f () = on the closed interval [-7,7]. Of the following intervals, on which can the Mean Value Theorem be applied to f? 11-1, 3 because f is continuous on (-1,3] and differentiable on (-1,3). II. [5, 7 because f is continuous on 5,7] and differentiable on (5,7). III. (1,5) because f is continuous on (1,5) and differentiable on (1,5). None © anal only
Let f:D + R be a function. (a) Recall the definition that f is uniformly continuous...
Let f:D + R be a function. (a) Recall the definition that f is uniformly continuous on D. (You do not need to write this down. This only serves as a hint for next parts.) (b) Use (a) and the mean value theorem to prove f(x) = e-% + sin x is uniformly continuous on (0, +00). (c) Use the negation of (a) to prove f(x) = x2 is not uniformly continuous on (0,0).
Let f be a differentiable function on R. Assume f' is continuous and always positive. You...
Let f be a differentiable function on R. Assume f' is continuous and always positive. You are searching for a root of f using Newton's method (see Tutorial 5). Your first guess is Xo ER and you compute subsequent guesses as follows: In EN, 2n+1 = In - f(2n) f'(x Let & E R. Prove that IF {Xn}"-o converges to & THEN x is a root of f.
Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some z...
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...
3. Let f: RP-R (a) If f(x)-Ax + b, x E R A є Mq.p and b є R9, show that f is p. where differentiable everywhere and calculate its total derivative (b) If f is differentiable everywhere and Df (x)A, for some A E Mp and all q.p x E Rp, show that there exists b E R, such that f(x) = Ax + b for all x E Rp
3. Let f: RP-R (a) If f(x)-Ax + b,...
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1) Show that the inverse function f -1 exists. (2) Prove that f is an open map (in the relative topology on I) (3) Prove that f1 is continuous
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1)...
23. Let be a function defined and continuous on the closed interval (a,b). If f has a relative maximum at cand a<c<b, which of the following statements must be true? 1. f'(c) exists. II. If f'(c) exists, then f'(c)= 0. III. If f'(c) exists, then f"(c)<0. (A) II only (B) III only (C) I and II only (D) I and III only (E) II and III only
Question 4* (Similar to 18.1) Suppose f is a continuous function on a closed interval [a, b]. In class, we proved that f attains its maximum on that interval, i.e. there exists Imar E la, so that f(Imar) > f(x) for all r E (a,b]. We didn't prove that f attains its minimum on the interval, but I claimed that the proof is similar. In fact, you can use the fact that f attains its maximum on any closed interval...
please explain in detail
4 -11 23 4 Graph of f Let f be a continuous function defined on the closed interval -1Sxs4. The graph of f, consisting of three line segments, is shown above. Let g be the function defined by g(x) = 5 +1.f(t) dt for-1 $154. (A) Find g(4). (B) On what intervals is gincreasing? Justify your answer. (C) On the closed interval 1 s xs 4, find the absolute minimum value of g and find the...
4. The function f is continuous on the closed interval (-2, 1). Some values of f are shown in the table below. --2 f(x) -3 -1 0 1 7 k3 The equation f(x) = 3 must have at least two solutions in the interval [-1,1) if k = a. 1 b. C. 2 CONN NICO d. 5. If k(r) is a continuous function over the interval (-2, 4) such that k(-2) = 3 and k(4) = 1, then k(2) 0...
Let f be the function given by f () = on the closed interval [-7,7]. Of the following intervals, on which can the Mean Value Theorem be applied to f? 11-1, 3 because f is continuous on (-1,3] and differentiable on (-1,3). II. [5, 7 because f is continuous on 5,7] and differentiable on (5,7). III. (1,5) because f is continuous on (1,5) and differentiable on (1,5). None © anal only
Let f:D + R be a function. (a) Recall the definition that f is uniformly continuous on D. (You do not need to write this down. This only serves as a hint for next parts.) (b) Use (a) and the mean value theorem to prove f(x) = e-% + sin x is uniformly continuous on (0, +00). (c) Use the negation of (a) to prove f(x) = x2 is not uniformly continuous on (0,0).
Let f be a differentiable function on R. Assume f' is continuous and always positive. You are searching for a root of f using Newton's method (see Tutorial 5). Your first guess is Xo ER and you compute subsequent guesses as follows: In EN, 2n+1 = In - f(2n) f'(x Let & E R. Prove that IF {Xn}"-o converges to & THEN x is a root of f.
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...
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