function f is differentiable on [a,b]and f(b)<f(a). Show that f'<0 at some point between a and...
(4) Let f(x) (0 if x<0 (a) Show that f is differentiable at z (b) Is f'continuous on R? Is f continuous on R? Justify your answer.
Let g be a function differentiable on R. Let x1 and x2 be a pair of numbers (x1 < x2) with the property that g(x1) = x2 and g(x2) = X1. Show that there exists a number where the value of g' is -1. Name any important theorem that you use.
(1 point) Consider the function f(x) = xe-5x, 0<x< 2. This function has an absolute minimum value equal to: which is attained at x = and an absolute maximum value equal to: 1/(5e) which is attained at x =
Convex Optimization Let f: R R be a differentiable function on R. Show that f is convex iff f' is nondecreasing (i.e. x y f'(x) <f'(y)).
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.
Determine if the following piecewise defined function is differentiable at x = 0. x20 f(x) = 4x-2, x2 + 4x-2, x<0 What is the right-hand derivative of the given function? f(0+h)-f(0) lim (Type an integer or a simplified fraction. I h h+0+
It is known that f :(0,2) + R is a differentiable function such that \f'(x) < 5 for all x € (0,2). Now let bn := f(2 – †) for all n € N. Prove that this is a Cauchy sequence.
Question 6 (1 point) Suppose a function f(x) is differentiable everywhere and has a local minimum at x=c. If f(x)<O when x<c, and f'(x)>0 when x>c, then by the Global Interval Method we know x=c is O a local maximum an absolute maximum a local minimum an absolute minimum
It is known that f :(0,2) + R is a differentiable function such that \f'(x) < 5 for all x € (0,2). Now let bn := f(2 – †) for all n € N. Prove that this is a Cauchy sequence.
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