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3. (25 pts) Suppose f(x) is twice continuously differentiable for all r, and f"(x) > 0...
Assume f : R" → R is twice continuously differentiable. Prove that the following are equivalent: ...
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...
linear optimization Assume that f : D → R is twice continuously differentiable for all x...
linear optimization
Assume that f : D → R is twice continuously differentiable for all x D, where the domain D off is an open, convex subset of Rn. Sh ▽2f(x), is symmetric positive-semi-definite for all x E D if and only if f is a convex function on D Moreover, if its Hessian matrix. ▽2 (x), is symmetric positive-definite for all x E D, then f is a strictly convex function on D Show that the converse of this...
Q3 (Prove that P∞ k=1 1/kr < ∞ if r > 1) . Let f : (0,∞) → R be a twice differentiable function with f ''(x) ≥ 0 for all x ∈ (0,∞). (a) Show that f '(k) ≤ f(k + 1) − f(k) ≤ f '(...
Q3 (Prove that P∞ k=1 1/kr < ∞ if r > 1) . Let
f : (0,∞) → R be a twice differentiable function with f ''(x) ≥ 0
for all x ∈ (0,∞).
(a) Show that f '(k) ≤ f(k + 1) − f(k) ≤ f '(k + 1) for all k ∈
N.
(b) Use (a), show that Xn−1 k=1 f '(k) ≤ f(n) − f(1) ≤ Xn k=2 f
'(k).
(c) Let r > 1. By finding...
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.
1 Let f: R R be a continuously differentiable map satisfying ilf(x)-FG) ll 리1x-vil, f Rn. Then fis onto 2. f(RT) is a closed subset of R'" 3, f(R") is an open subset of RT 4. f(0)...
1 Let f: R R be a continuously differentiable map satisfying ilf(x)-FG) ll 리1x-vil, f Rn. Then fis onto 2. f(RT) is a closed subset of R'" 3, f(R") is an open subset of RT 4. f(0)0 or all x, y E 5) S= (xe(-1,4] Sin(x) > 0). Let of the following is true? I. inf (S).< 0 2. sup (S) does not exist Which . sup (S) π ,' inf (S) = π/2
1 Let f: R R be...
13. Write a MATLAB program to find all the roots of a given, twice continuously differentiable,...
13. Write a MATLAB program to find all the roots of a given, twice continuously differentiable, function f e C2la,b]. on the given interval to find out where it Your program should first probe the function f(x) changes sign. (Thus, the program has, in addition to f itself, four other input arguments: a, b, the number nprobe of equidistant values between a and b at which f is probed, and a tolerance tol.) For each subinterval [a,b;] over which the...
4 Suppose f : (0,0) → (0,x), is a differentiable function satisfying f(a +b)-f(a)fb), for all...
4 Suppose f : (0,0) → (0,x), is a differentiable function satisfying f(a +b)-f(a)fb), for all a,b>0 Moreover, assume that f(0)1 (a) Prove that there exists λ (not necessarily positive) such that f(r) = e-Ar, for all r. Hint Find and solve a proper differential equation. (b) Suppose that X is a continuous random variable, with P(X>ab)-P(>a)P(X> b), for all a, b e (0, oo). Prove that X is exponentially distributed
Suppose that f is twice differentiable function where f(0)=f(1)=0. Prove that strategy Suppose that f is a twice differ...
Suppose that f is twice differentiable function where
f(0)=f(1)=0.
Prove that
strategy Suppose that f is a twice differentiable function where f(0) = f(1) = 0. 1 Prove that f f"(x)f (x) dx a. Using part a, show that if f"(x) = wf (x) for some constant w, then w 0. Can you think of a function that satisfies these conditions for some nonzero w? b.
strategy Suppose that f is a twice differentiable function where f(0) = f(1) =...
(25 pts) For f(x) infinitely continuously differentiable, and C so that the formula Taylor series to...
(25 pts) For f(x) infinitely continuously differentiable, and C so that the formula Taylor series to find A,B, use Af (x 2h)Bf(x) Cf (xh) gives the highest order accurate approximation of f'(x) (for general f and x). What is that order? Remember, Taylor series says h2 |f" (x)^f"(x) h3 f(xh) f(x)hf'(x)+ ... 2! 3!
(25 pts) For f(x) infinitely continuously differentiable, and C so that the formula Taylor series to find A,B, use Af (x 2h)Bf(x) Cf (xh) gives the...
9. Suppose that f : [0,-) + R is differentiable and that the derivative f' :...
9. Suppose that f : [0,-) + R is differentiable and that the derivative f' : [0,00) + R is also differentiable, with f(0) = f'(0) = 0. Suppose also that [f"(x) < 1 for all € [0, 0). a) Show how the Mean Value Theorem can be used to prove that f(x) <r? for all x € (0,00). b) Show how the Cauchy Generalized MVT can be used to prove a stronger statement: |f(7) < 2 for all 2...
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...
linear optimization
Assume that f : D → R is twice continuously differentiable for all x D, where the domain D off is an open, convex subset of Rn. Sh ▽2f(x), is symmetric positive-semi-definite for all x E D if and only if f is a convex function on D Moreover, if its Hessian matrix. ▽2 (x), is symmetric positive-definite for all x E D, then f is a strictly convex function on D Show that the converse of this...
Q3 (Prove that P∞ k=1 1/kr < ∞ if r > 1) . Let
f : (0,∞) → R be a twice differentiable function with f ''(x) ≥ 0
for all x ∈ (0,∞).
(a) Show that f '(k) ≤ f(k + 1) − f(k) ≤ f '(k + 1) for all k ∈
N.
(b) Use (a), show that Xn−1 k=1 f '(k) ≤ f(n) − f(1) ≤ Xn k=2 f
'(k).
(c) Let r > 1. By finding...
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.
1 Let f: R R be a continuously differentiable map satisfying ilf(x)-FG) ll 리1x-vil, f Rn. Then fis onto 2. f(RT) is a closed subset of R'" 3, f(R") is an open subset of RT 4. f(0)0 or all x, y E 5) S= (xe(-1,4] Sin(x) > 0). Let of the following is true? I. inf (S).< 0 2. sup (S) does not exist Which . sup (S) π ,' inf (S) = π/2
1 Let f: R R be...
13. Write a MATLAB program to find all the roots of a given, twice continuously differentiable, function f e C2la,b]. on the given interval to find out where it Your program should first probe the function f(x) changes sign. (Thus, the program has, in addition to f itself, four other input arguments: a, b, the number nprobe of equidistant values between a and b at which f is probed, and a tolerance tol.) For each subinterval [a,b;] over which the...
4 Suppose f : (0,0) → (0,x), is a differentiable function satisfying f(a +b)-f(a)fb), for all a,b>0 Moreover, assume that f(0)1 (a) Prove that there exists λ (not necessarily positive) such that f(r) = e-Ar, for all r. Hint Find and solve a proper differential equation. (b) Suppose that X is a continuous random variable, with P(X>ab)-P(>a)P(X> b), for all a, b e (0, oo). Prove that X is exponentially distributed
Suppose that f is twice differentiable function where
f(0)=f(1)=0.
Prove that
strategy Suppose that f is a twice differentiable function where f(0) = f(1) = 0. 1 Prove that f f"(x)f (x) dx a. Using part a, show that if f"(x) = wf (x) for some constant w, then w 0. Can you think of a function that satisfies these conditions for some nonzero w? b.
strategy Suppose that f is a twice differentiable function where f(0) = f(1) =...
(25 pts) For f(x) infinitely continuously differentiable, and C so that the formula Taylor series to find A,B, use Af (x 2h)Bf(x) Cf (xh) gives the highest order accurate approximation of f'(x) (for general f and x). What is that order? Remember, Taylor series says h2 |f" (x)^f"(x) h3 f(xh) f(x)hf'(x)+ ... 2! 3!
(25 pts) For f(x) infinitely continuously differentiable, and C so that the formula Taylor series to find A,B, use Af (x 2h)Bf(x) Cf (xh) gives the...
9. Suppose that f : [0,-) + R is differentiable and that the derivative f' : [0,00) + R is also differentiable, with f(0) = f'(0) = 0. Suppose also that [f"(x) < 1 for all € [0, 0). a) Show how the Mean Value Theorem can be used to prove that f(x) <r? for all x € (0,00). b) Show how the Cauchy Generalized MVT can be used to prove a stronger statement: |f(7) < 2 for all 2...
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