For the function f(x), determine whether or not f is continuous and/or differentiable at the following...
a. Determine whether the Mean Value Theorem applies to the function f(x) = x + on the interval [3,5). b. If so, find or approximate the point(s) that are guaranteed to exist by the Mean Value Theorem. a. Choose the correct answer below. O A. No, because the function is continuous on the interval [3,5), but is not differentiable on the interval (3,5). OB. No, because the function is differentiable on the interval (3,5), but is not continuous on the...
5. 6 pt Determine whether the function f(x) is continuous and/or differentiable at x = 1. (x2+1 f(x) = { 12, >1 1 <1
part a and b a. Determine whether the Mean Value Theorem applies to the function f(x) x+ on the interval(-4,-3) b. If so, find or approximate the point(s) that are guaranteed to exist by the Mean Value Theorem a. Choose the correct answer below O A. No, because the function is not continuous on the interval (-4,-3), and is not differentiable on the interval (-4,-3). OB. No, because the function is differentiable on the interval (-4,-3), but is not continuous...
5. 5 Let f(x) be continuous and differentiable on f(5) = 4. Mark TRUE or FALSE for the following statements and JUSTIFY. (No points will be given without the correct justification) [0, 10] with f(0) = f(10) 0 and (E) There is some c E (0,5) such that f'(c) = 5. 5 Let f(x) be continuous and differentiable on f(5) = 4. Mark TRUE or FALSE for the following statements and JUSTIFY. (No points will be given without the correct...
12x + 24 12 if r 21 (a) To determine whether f is differentiable atェ=-1, we compute linn f(-1+h)-/(-1 h-+0+ linn f(-1+h)-f(-1) = linn h-40- (b) To determine whether fis differentiable at r-0we compute lim = INF (c) To determine whether f is differentiable at r 1, we compute -40% (1 +h)-10) lim lim linii (d) Classity the differentiability of f at each of the points f has a cusp at x-a ( f has a vertical tangent line at...
At x = 1 the function (in the graph below) is o continuous, differentiable o continuous, not differentiable o not continuous, not differentiable o not continuous, differentiable
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) =
33. (a) Use the following graph of f(x) to determine whether f'(x)exists at x=a, b,andc. A (b) Let g(x) = Vx-5. A student graphed this function, and the graph appeared to be smooth and continuous for all real numbers x. The student concluded that g(x) is differentiable for all x, which is false. Identify the error, and explain why the student's conclusion is false. What is the correct conclusion regarding the differentiability of g(x)?
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,...
(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.