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a. The function f: RR given by f(x)=x² is twice continuously differentiable in R", and its Hessian is v² f(x) = VxER".

a. The function f: RR given by f(x)=x² is twice continuously differentiable in R", and its Hessian is v² f(x) = VxER".

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Which of the following statements are true?

 

a. The function f: R→R given by f(x)=12||x||² is twice continuously differentiable in R", and its Hessian is exists. In this case, we write c. Every differentiable function is a polynomial. Od. We say that a function f: R→R is differentiable at x in R if there exists a in R such that Of. b. We say that a function f: R→R is differentiable at x in R if for every sequence (xk) in R with limk-ooXk-X and xk*x for all k in N, the limit f(xk) - f(x) xk x In this case, we write f'(x)=a. ☐e. We say that a function f: R→R is differentiable at x in R if the limit exists. In this case, we write v² f(x) = The function f: R→R given by f(x)=½||x||2 is continuously differentiable in R", and its gradient is lim k→∞o lim 0#h→0 |h| 3 f'(x) = lim lim 0#h→0 ⠀ VxER". -\f(x +h) − f(x) – ah| = 0. f(xk) - f(x) k→∞o Χk - X f'(x) = lim 0#h→0 f(x +h)-f(x) h f(x+h)-f(x) h Vf(x)=xVxER".
Transcribed Image Text:a. The function f: R→R given by f(x)=12||x||² is twice continuously differentiable in R", and its Hessian is exists. In this case, we write c. Every differentiable function is a polynomial. Od. We say that a function f: R→R is differentiable at x in R if there exists a in R such that Of. b. We say that a function f: R→R is differentiable at x in R if for every sequence (xk) in R with limk-ooXk-X and xk*x for all k in N, the limit f(xk) - f(x) xk x In this case, we write f'(x)=a. ☐e. We say that a function f: R→R is differentiable at x in R if the limit exists. In this case, we write v² f(x) = The function f: R→R given by f(x)=½||x||2 is continuously differentiable in R", and its gradient is lim k→∞o lim 0#h→0 |h| 3 f'(x) = lim lim 0#h→0 ⠀ VxER". -\f(x +h) − f(x) – ah| = 0. f(xk) - f(x) k→∞o Χk - X f'(x) = lim 0#h→0 f(x +h)-f(x) h f(x+h)-f(x) h Vf(x)=xVxER".
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