Please explain in complete details!
Please explain in complete details! A function f is continuous on [a, b] and differentiable on...
4. [-/1 Points] DETAILS LARCALC11 3.2.006. Explain why Rolle's Theorem does not apply to the function even though there exist a and b such that fe 8(+) = N(2 – 2+)'. 1.1,1) None of these. f(a) does not equal f(b) for all possible values of a and b in the interval [-1, 1). Fla) does not equal f(b) for any values in the interval [-1, 1). There are points on the interval [a, b] where fis not continuous. There are...
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,...
Let f: a, b R be a function, continuous on a, b and differentiable on (a, b). Show that 3c E (a, b) such that f (b) f(c) <0 f (e) f(a) 3s E (a, b) s.t f'(s) 0.
For the function f(x), determine whether or not f is continuous and/or differentiable at the following points. Also using only the given function (not a graph), determine what occurs graphically at these points. f(x) = 1, X, x² - 12, x < 0 0<x< 4 X > 4 (a) At x = 0, f(x) is ---Select--- . At this point, the graph of f(x) has ---Select--- (b) At x = 2, f(x) is ---Select--- . At this point, the graph...
Suppose f is a continuous and differentiable function on [0,1] and f(0)= f(1). Let a E (0, 1). Suppose Vr,y(0,1) IF f'(x) 0 and f'(y) ±0 THEN f'(x) af'(y) Show that there is exactly f(ax) and f'(x) 0 such that f(x) one Hint: Suppose f(x) is a continuous function on [0, 1] and f(0) x € (0, 1) such that f(x) = f(ax) f(1). Let a e (0,1), there exists an Suppose f is a continuous and differentiable function on...
7.(a) Show that the following function is continuous at c 0 but is not differentiable at c = 0: x cos), ifx 0 f(x) 0, ifx 0 (b) Show that the following function is differentiable at c 0: (x2 cos ), ifx 0 0, if x 0 (x) =
2. Rolle's theorem states that if F : [a, b] → R is a continuous function, differentiable on Ja, bl, and F(a) = F(b) then there exists a cela, b[ such that F"(c) = 0. (a) Suppose g : [a, b] → R is a continuous function, differentiable on ja, bl, with the property that (c) +0 for all cela, b[. Using Rolle's theorem, show that g(a) + g(b). [6 Marks] (b) Now, with g still as in part (a),...
please explain your solution with details. 7) Let f and g be differentiable functions such that 2< f(x)<4 and 2 s g(x)< 4 for all x. a) Find good upper and lower bounds on the arc O to x 4.(5 pts) length of the graph of f(x) from x= Ax,2 494, ANS L4 . Are Cang th Say tnt 1 + We can b) Can we find a good lower bound on the length of g(x) from x 0 to...
Consider the function f(x) = Σ (a) Where is f defined? (b) Where is f continuous? (c) Where is f differentiable? Consider the function f(x) = Σ (a) Where is f defined? (b) Where is f continuous? (c) Where is f differentiable?
1. Consider the function f(a) 2-9 3, but not differentiable at z Show that f( ) is continuous at r- 3.