4. Given a function f that is differentiable on (a, b), is it possible for there...
(1 point) For the function f(x) = e2x + e- defined on the interval (-4, o), find all intervals where the function is strictly increasing or strictly decreasing. Your intervals should be as large as possible. f is strictly increasing on f is strictly decreasing on (Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10)) whenever r is near c on the left Find and classify all local max's and min's. (For the purposes...
5. Consider the function f: R -> R given by f (x, y) := e°+v* _ 4. (a) Sketch the level curves of f. (5 marks) (b) Find Vf, the gradient of f, and determine at which points Vf is zero. Remark: These points are called the critical points of f (5 marks) (c) Determine whether the critical points of f are local minima, local maxima, or saddle points by considering the level curves of f. (5 marks) (d) Calculate...
Find all the local maxima, local minima, and saddle points of the given function.f(x,y)=x²+xy+y²+6x-6y+7Select the correct choice below and fill in any answer boxes within your choice.A. There are local maxima located atB. There are no local maxima.A. There are local minima located atB. There are no local minima.A. There are saddle points located at
4. For this question, define f(x) = (x - 1)e-(0-1). (a) Find f'(x) and f"(x). (b) Find where S is increasing and where / is decreasing (e) Find where S is concave up and where / is concave down. (a) Find all critical points of . For each point you find, explain whether it is a (relative) maximum, a (relative) minimum or neither. (e) Find all points of inflection of f. For each point you find, explain why it is...
1) Suppose f (a, b) R is continuous. The Carathéodory Theorem says that f(x) is differentiable at -cE (a, b) if 3 (a, b)-R which is continuous, and so that, (a) Show, for any constant a and continuous function (x), that af(x) is continuous at z-c by finding a Carathéodory function Paf(x). (b) Show, for any constants a, B, that if g : (a, b) -R is differentiable at c, with Carathéodory function pg(z), then the linear combination of functions,...
function f is differentiable on [a,b]and f(b)<f(a). Show that f'<0 at some point between a and b.
Let f be a twice differentiable function on an open interval (a, b). Which statements regarding the second derivative and concavity are true? If f"(c) is positive, then the graph of f has a local maximum at x = c. The concavity of a graph changes at an inflection point. If f is increasing, then the graph of f is concave down. The graph of f has a local minimum at x = c if f"(c) = 0. The graph of f is concave up if...
1. Give an example of a differentiable function f and a point xo in the domain of f such that f(xo) # Poo(xo), where Poo is the Taylor series of f centered at x = 1. (To be perfectly precise, f(x0) + P(xo) means that lim En(xo) = 0, where En(xo) is the usual error function evaluated at xo.) n- 00 extex 2. The function cosh(x) = = - is called 2 the hyperbolic cosine and has many applications in...
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.
constraint* is mispelled f(x, y) 2x2 -12xy2- 6y 10o a) Explore the function for local minima and maxima: find critical points and determine the b) Explore the given function for absolute maximum in the closed region bounded by the type of extremum triangle with vertices (0,0), (0,3) and (1,3) Explore the function at each of three borders. Determine absolute maximum and minimum c) Find critical points of the given function f(x, y) under the constrain xr_y2x = 4x + 10...