Let –1, -T<¤ < 0, 0, 0 < x < T. f(x) = * = 0 1,

Q: 4. (5pt) If f(r) = [t – 1) cosh t dt, on what interval is ƒ increasing? (Hint: Use the Fundamental…

A:

Q: Find an example of a function f:[0,1]¬R such that f is not continuous, but f does satisfy the…

A:

Q: 8 Examine holomorphism for given function f(z) = |z|

A: The given problem is to examine the holomorphism of the given complex function. Given function…

Q: Let f(x) = cos (x). %3D If it is true that f" (x) = x(f'(x))ª for some constant a, find the value of…

A:

Q: (5) Use the limit definition of derivative to prove the following: (a) For any a e R, (f(ax))' =…

A:

Q: '3 Suppose that f" is continuous on [1, 3] and f(1) = 8, f(3) = -3, f'(1) = 7, and f'(3) xf "(x) dx.…

A:

Q: 6. Consider the function f Cx)=5(o+5)-25 if X#し if xfO D.IS fCx) continuous at X-0? EXPlain

A:

Q: 16. Suppose that f is differentiable at c and that f'(c) + 0. Show that there exists a ð > 0 such…

A:

Q: 3. Let f(x) = x, given the interval [0,1]

A: fx=x, x∈0,1 f'x=1&gt;0 for all x∈0,1 So, fx is strictly increasing on the interval 0,1.

Q: Suppose that ƒ is continuous in [0,1] and that f(x) E [0, 1] for all x. Show that f(x) = x for some…

A: Given that f:0,1→0,1 is continuous on 0,1. To show fx=x for some x∈0,1

Q: of of and Find for the following function. f(x.y) = e xy In (2y) of dy

A:

Q: If f(x) is a continuous function on [3,7] and F(x) is an anti-derivative of f(x) such that F(3)=7…

A: Given :f(x) is continuous function 3,7Fx is anti-derivative of fxF3=7 and F7=3To determine…

Q: 2. Let f(x) = S dt for x 2 2 In a) Find F'(x). b) Is F increasing or decreasing? What can you say…

A:

Q: 4. Let f(z) = cosh z. (3) Show that f: S→ f(S) is ono-to-one.

A:

Q: 1 Let f(x) = 8 + In (t) dt, for x2 1. Find f'(2).

A:

Q: 3 Suppose that e(x) and f(x) are differentiable on [0, 1] and that 0 < |e'(x)| < |f'(x)| for all x €…

A:

Q: 1 - t Let F(x) = dt for -o << x < +∞ t2 + 24

A:

Q: If f is continuous on [-1, 1] and f(x)dx = 0, then there exists some ce [-1,1] for which f(c) = 0.

A: Given: f is a continuous function on the interval -1, 1 and ∫-11f(x) dx=0. Prove: There exists some…

Q: [0, 4). 4. Prove that f(x) = x² is both uniformly continuous and continuous on

A:

Q: 1 Let f (x) = ; 1 , then f (x) satisfies the - - 2 conclusion of Rolle's Theorem on 0, 1| 6.

A:

Q: Suppose that the function f: [0, 1]  in R is continuous and thatf (x)&gt;= 2     if 0 &lt;= x &lt;…

A:

Q: Suppose that a function h satisfies −x^2 ≤ h(x) ≤ x^2 for all x. Prove that h is differentiable at x…

A:

Q: Let A > 0 be a fixed but unspecified constant, and let f(x) = xn= (In x)", x > 1. Compute f'(x).

A: our objective is to find the derivative

Q: Find the values of α and β such that the function f(x) = ( α cos x + 2 if x < 0 βe3x + αx2 if x ≥ 0…

A:

Q: X if x e Q If f (x) is defined on [0,1] by the rule f (x) = < (1-x , if x 4 Q Prove that fof (x) = x…

A:

Q: 9. Consider the function f: R → R, defined by (x²sin(1/x) ifx # 0 f(x) = if x = 0 Then a. f is not…

A:

Q: Assume that ƒ is continuous on [0,1] and that f(0) = 2 and f(1) = 1. Show that the graph of y = f(x)…

A: given  f is continuous on [0,1] f0=2,f1=1y=x2+1

Q: Use the e- 8 property in Definition 19.1 to show that f(x) 1 is uniformly continuous on the interval…

A:

Q: 1 If f(x) is a non-negative continuous function for all x, such that f(x) + + f(x+1/2) = 1,0 ≤ x ≤ -…

A: Given that fx is a non-negative continuous function for all x, such that fx+fx+12 = 1 for 0 ≤ x ≤…

Q: Prove, if f(x)= V1+x² –In +1+ 1 is increasing and concave on x X E (0;+∞) .

A: f is increasing if first derivative is positive in the interval  and f has concave if second…

Q: Find the value of a such that the function S(«) = {** -1, x < 3 x 2 3 | f(x) = 2ах, is continuous at…

A: Given:      f(x) = x2 - 1  , x &lt; 3     f(x) = 2ax  ,   x ≥ 3

Q: 4. If continuous function f:[0,1] U (2,3) → R is one-to-one, is f also continuous? 1

A:

Q: 3. Define f(2) in a way that extends f(s) = -8 to be continuous at s = 2. s2 – 4

A:

Q: The graph of f(x)= x2 -1 is concave up on interval (-1,1)^O (0,1) 0 (-1,0) U (1, 0 )co…

A: Given,  The function fx = xx2-1  We have to find the interval in which the graph of  fx = xx2-1 is…

Q: Let fn f, where if 0<<! Sn(x) ={ S 2-a if 0<x<1 and f(x) if a = 0 Then lim sup fn (x) – ƒ (x)| =

A:

Q: (c) Suppose that f : [-1,1] →R is a continuous function such that r²+ f(x)² = 1 for every rE[-1, 1].…

A: Given that the domain of the given function is [-1, 1] and function is defined as: f:−1,1→ℝ​is a…

Q: Show (in terms of  −δ) that a function f : [2,7] −→R be defined by f(x) = √x2+1 is uniformly…

A:

Q: 1. Determine whether k(x) is continuous at r = 3, where %3D x + 3, x 2.

A: Given  k(x)=x+3,  x&lt;0x2+1, 0≤x≤22x,   x&gt;2 To check the continuity we can use the graph as…

Q: 1 dt, for x > 4. In(t) Let F(x) 4

A:

Q: 1. Show (in terms of -5) that a function f: [2, 7] R be defined by f(r) V+1 is uniformly continuous.

A:

Q: Let f be a differentiable function with f(0) = 1 and / f'(1) dt = x². Then Of(x) = COS X Of(x) = (x+…

A:

Q: 3 - x, x<2 Q.No.15: Is f(x) = {+2 x22 continuous at every x € R. x+1 xs1 -.

A: Continuity of a function.

Q: Determine if f(x)=In (x² + 1) is uniformly continous on the interval I (-0, +0)

A:

Q: Suppose f is a differentiable function on an interval II and f(x) 6= 0 for every x in the interval.…

A:

Q: 1. Suppose f(r) is a continuous function on r € [a, b]. Prove f(x)dr < S°S(x)| dx.

A:

Q: If ø(x) =(x)+ f(1-x) and f"(x)<0 for all x e increasing in [0,1/2] and decreasing in (1/2,1].…

A:

Q: 2. Suppose that f is a measurable function satisfying m{r ER: |f (x)| > n} ) < 1 nEN. n3 Is f…

A:

Q: Prove that there is no continuous function f : S² → S' such that f(-x) =-f(x) for all x € S².

A: