8. Given the following function f, define f(1) so that f is con- tinuous at x = 1, if possible: Зх — 1, ifx < 1 |x² + 1, if x > 1. f(x) =

Q: The following statement is false, explain why. The function g(x) = x^2 on the interval [-1,1] has…

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Q: Suppose that f: [0,1]–R is a function satisfying the conclusions of the intermediate value theorem.…

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Q: 3. Show that the functionf on [-1,1] defined as if f (r) = 1+e if x = 0 is discontinuous at x = = 0.…

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Q: + [ƒ(2)₁ Suppose that fand gare continuous functions such that 1 [ g(x)dz -3, find f(x)dx= 2 and +…

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Q: Suppose f(x) = - 2x - cosr. (a) Show that f is strictly decreasing on R. (b) Hence, justify that 2x…

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Q: 2. If f(x) = cos2x, find f(0), f(--).f),f(-x) 3. If f(x) = x(x+1), show that f(x+h)– f(x)= h(2x +1+…

A: 2. f(x) = cos2x , find f(0) , f(-π/4), f(7π/8) ,                                         f(-x). f(0)…

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

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Q: 1. Given the graph of f', the derivative of f, locate all extrema of f. JUSTIFY. Minimum at x =…

A: To find the minima and maxima from the graph of f'(x)

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

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Q: 10. Consider the following function. 1 x } x > Find the values of a and b that make f(x)…

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Q: Use Theorem 2 to show that 3, if x <-10 2, if – 10 sxso F:R → R,F(x) = -2, if 0< x< 10 -3, if x2 10…

A: so we have to show that given function is discontinuous at 2 and -2 (a)   Discontinuous  at -2…

Q: 18. Suppose that f is a differentiable function such that f(g(x)) = x, and f'(x) = 1 1+(f(x))². Show…

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Q: Suppose f(x) = – 2x – cos r. (a) Show that f is strictly decreasing on R. (b) Hence, justify that 2x…

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Q: 1. Given the function 2r + 1, r 1 Verify whether or not the function f is continuous at r 1.

A: the given function is: fx=2x+1,     x≤14x2-1,       x&gt;1 we have to verify whether or not the…

Q: I'm stuck on this question and I don't know how I should be approaching this. What should I do?

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Q: 1. Suppose f is a function which satisfies f(x + y) = f(x) + f (y) for all x,y e R. What is the…

A: Let f is a function which satisfies f(x+y) = f(x) + f(y)  for all x,y∈R   f(x+y) = f(x) + f(y) ,   ∀…

Q: 4 Given that f(2) is analytic at z = 0 and that f G) =, n = 1,2, 3..,find f(2). VE

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Q: Suppose that f and g are discontinuous at x = c. Does it follow that f + g is discontinuous at x =…

A: Let us define that Then both f and g are discontinuous at x=2

Q: If 7х- 5,х <1 { h(x) = 3x2 – x , 1 < x< 3 Is h(x) continuous at (x=1)?

A: Continuity of function at a Point : Suppose a function f:A→B where A,B⊆ℝ .Then f is continuous at…

Q: 45. For what value of a is x2 - 1, x < 3 2ax, f(x) = %3D 2ах, continuous at every x?

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Q: 4. Find the first two intervals using the bisection method to compute the minimum of the function…

A: We need to find the first two intervals of on the interval

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

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Q: The graph of f(x)= x2 - 1 is concave down on interval | (-1,1) ^O (0,1) BO (-1,0) U (1, ∞ )co (- 00…

A: From the given problem:   fx=xx2-1   As we know that for concave downward:   f''x&lt;0   So,…

Q: Can Rolle's Theorem be applied to f(x) = 1/(1-x) on interval [0,3]? If yes, what is c in (0,3) such…

A: here we can't apply the rolles Theorem

Q: (11) (C) For the function f(x) = x° – x, determine whether the hypothesis of Rolle's theorem hold on…

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Q: 2- For what value of the costant c is the function continuous at every number? F(x) = {x+c if x<2…

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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…

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Q: 1. Apply the Parseval's identity to the function f (x) = x, –n < x < n

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Q: 11. Prove from the definition that the following functions are differentiable. ()- (b) f =xy?

A: By the definition, the function f(x,y)=xy^2 is differentiable if fx(x,y) and fy(x,y) both exist.

Q: 5. Show that the function h(x, y) defined by 1 if a? + y? 1 is discontinuous.

A: We can solve this using R.H.L ≠ R.H.L

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

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Q: Suppose that f (0) = 2 and f' (x) ≤ 3 for x &gt; 0. Apply the MVT to the interval [0, 4] to prove…

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Q: Since the simplified numerator, >0 for all X, we have "(x) > 0 x²-1>0 e ]x > and f"(x) < 0 X| < Thus…

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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: Which of the following tatements is true about a mooth function f at (a, b) it ² (a, b) c, fax (a,…

A: f has relative maxima at (a, b) if D&gt;0, fxx(a,b)&lt;0 f has relative minima at (a, b) if D&gt;0,…

Q: 2. Consider the function f with domain (-2, 0): 3|22 – 1| if – 2<r<1 2.r2 + 2x – 4 f(x) = 3 if r = 1…

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Q: Let f = f(x, y), x = s +t and y = s - t. Show that (9- ) - se se as at ax ду

A: It is given that, f=f(x,y), x=s+t and y=s-t. Show that, ∂f∂x2−∂f∂y2=∂f∂s·∂f∂t.…

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

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Q: Find m so that f(x) is continuous at x = 2. x +4 if x 2 -

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Q: 9. If f(x) = ], ( 22 - 5t-3 ]dt, on what interval(s) 0, is f increasing, decreasing

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Q: 8. Use the c, 8 definition to show that fAx) = x? +1 is continuous at

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Q: For every continuous f:[0,+0) →R _function , if a>0 2fx'f(x*)dx =[ xf(x)dx Demonstrate that this…

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Q: 5. Consider the function S 2r – 1, si x < 2 f(x) = 6-x, si x 2 2. The Limit x 2 f(x) is:

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Q: Q.4 A/ Prove the function continuous at x = -2, (x2 + 5 13-3x where %3D if x<-2 if x 2-2 f(x) = %3D

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Q: 2. Find the (exact) values of a and b so that f is continuous at a = 2. 8r2 + 20x + 28 if x 2

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Q: Let f(x) = 2 cosh(4x) – e4x %3D Find the absolute maximum of f(x) on the interval [0, In 3].

A: The given function is                                         f(x)=2 cosh(4x)-e-4x We know that…

Q: 2. Consider the function f(x) = if x = {9 in lowest terms q > 0, p,q e Q; 0 if x e Q°,x = 0…

A: We use sequential approach of continuity to show it.

Q: Suppose f(x) is a bounded real - valued function on [a.b) such that fela,b] . Does it follows that…

A: A function is bounded in [a,b] if it can't achieve value beyond a and b. i.e.maximum acheivable…

Q: Suppose f is a function such that f(z) = 4z' and f" (z) = 12z. Which of the following is TRUE? f has…

A: Suppose f is a function such that f'(x)=4x3  &amp;  f"(x)=12x2

Q: .For what values of a is Jaax - 2a, x z 2 f(x) 12, x < 2 continuous at every x?

A: Here given function  Since,  it is continuous for every x thus it Is also continuous for x=2