8. Constantly Differentiable continuation
Determine a function f: R->R that apply to the following properties
- For all applies f(x) = sin(x)
- For all ,applies f(x) =
- f is continuously differentiable
8. Constantly Differentiable continuation Determine a function f: R->R that apply to the following properties - For a...
8. Constantly Differentiable continuation Determine a function f: R->R that apply to the following properties - For all applies f(x) = sin(x) - For all ,applies f(x) = - f is continuously differentiable r e-oo, 0 OC e1, o0) We were unable to transcribe this image
Can you find a differentiable function f(x) defined on the interval [0, 3] such that , and for all x ∈ [0, 3]? Justify your answer (do not write only Yes or No, but explain your answer). We were unable to transcribe this imageWe were unable to transcribe this imagef'(x) <1
A scalar function f : which is never zero has the properties and Evaluate the integral where is the surface of the unit sphere and means the directional derivative of f in the direction of the outward pointing unit normal on . R? → R We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe...
Let ⊂ be a rectangle and let f be a function which is integrable on R. Prove that the graph of f, G(f) := {(x, f(x)) ∈ : x ∈ }, is a Jordan region and that it has volume 0 (as a subset of ). We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
4. True or False. Write true or false in the blanks. a, A continuous function over a closed interval will achieve exactly one local maximum on that interval ______________ b. If f(x) and g(x) both have a local maximum at x=a then has either a local maximum or a local minimum at x=a. ___________ c. If for all x and if a > b, then _____________ d. If is undefined, and if is continuous at x=c, then has a local...
1.1 Be f: R->R given by , Show that f ist convex 1.2 Be f: given by . Show that f ist convex 1.3 Show, that for all applies : f(x) (0.00) → R Oo f(r) =-In(2) We were unable to transcribe this imageInla f(x) (0.00) → R Oo f(r) =-In(2) Inla
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...
differentiable function and there exists 0 <A < 1 (6) Suppose that f : R" -> R" is a such that |f'(x)|< A, for all x E R". Prove that the function F(x)= x - f(x) maps R" one-to-one and onto R". (Suggestion: Use the Contraction Mapping Principle Why not use the Inverse Function Theorem?) differentiable function and there exists 0
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...
Note that f is twice differentiable on means that the second derivative f''(x) exists for each x that is an element of Explain in full details! Suppose f is a twice-differentiable function on (0,00) (i.e. f"(x) exists for every 1 € (0,0)), and A = sup \f(x)] = sup{]f(x)]: 0<x<00}, B = sup \f" (x)], C = sup 18" (2)]. O<I<0 0<<< O<I<00 If A+C< show that BP < 4AC. (0,00 We were unable to transcribe this image