et f : [0, 1] → R be a continuous function on the closed interval [0, 1] and differentiable on the open interval (0, 1). Assume that f(0) = 0 and f’ is an increasing function on (0, 1). Show that g(x) =f(x)/x is an increasing function on (0, 1)

Q: Prove there does not exist a continuous function which maps the closed interval [0, 1] onto the open…

A:

Q: Sketch the graph of f(x) if ...f'(x)>0 on (2,4) and f'(x)<0 on (−∞, 2) ∪ (4, ∞),f''(x)>0 on…

A: Here we suppose that the given function is continuous on the given intervals and differentiable on…

Q: Let f(x) be a function with domain (-00, 00) and f'(x) = the intervals where f is decreasing. OA B…

A:

Q: Find continuous functions f1, f2, f3, f4 defined on the open unit interval I = (0, 1) (i.e.…

A: 1. f1(x) = (M1 - L1)x + L1, for all x in (0, 1)  This function is linear and maps (0, 1) onto (L1,…

Q: Let f be a function defined on [15, 23]. Suppose for every x ≤ (15, 23) d da If f(16) = 44, find the…

A: We  know By definition of derivative  in interval a,b⇒f'x=limx→afx-fax-a

Q: Let f be a function that is continuous on [1,7] and differentiable on (1,7). Suppose 2 < f'(x) < 3…

A:

Q: Suppose we have a function f on an interval [a,b] where f(a)=f(b)=0 and f''(x) <= 0 for all x on the…

A:

Q: Construct a continuous function that shows that the intervals [0,1] and [2,6] are homeomorphic

A: In the given question we have to find a continuous function from the interval [0,1] to the interval…

Q: Select the graph of a function f that satisfies f' 0 on (-0, -1)U(-1, 0).

A: means f is decreasing for x&gt; - 1   means f(x) is concave up on both sides of x=-1

Q: A function f: R → R has a third order continuous derivative. Show that f(a) f'(a). f"(a). f"" (a) ≥…

A: Given f:ℝ→ℝ has a third order continuous derivative Case 1: Consider fx is even function f-x=fx…

Q: Let f and h be real-valued functions continuous on [a, b], differentiable on (a, b), and h(a) not…

A:

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.