Show that the function f (x) = (x – a) (x – b)² + x | takes on the value (a + b) /2 for some value of x. (Hint: assume a <

Q: Find c such that the function x2 – 9 x с is continuous everywhere. - С —

A: f(x) will be continuous at x=c. when value of the function at c , left hand limit of f(x) at c  and…

Q: ): Apply Rolle's theorem for the function f(x) = 9x² – 36x + 27 over the interval [1, 3]. Justify…

A: Given :- Apply Rolle's Theorem for the function fx = 9x2 -36x + 27 over the integral 1,3.

Q: he domain of the function f(x) = Vx+5 is In(3-x) %3D

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Q: Find the value of the constant (a , b, or c) that makes the function continuous.

A: If function is continuous  Then, it will be continuous on x=-1 and x=12 Therefore, limx→-1-1fx=f-1…

Q: Suppose that f(x) is a differentiable function such that f'(4)=5 and f(4)=4. If g(x)=xf(x), then…

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Q: Find the domain of f(x) = 3x/(x − 1) and discuss the behavior of f near any excluded x-values.

A: Given function is f(x)=3xx-1 Since domain is the set of all those input values for which the…

Q: 4х+7 (ii) If f(x)={ 1<x<2 determine whether the function f(x) is continuous 14x² -1 2<x<4 throughout…

A: Continuity of the function

Q: 2. Determine whether the function y= -x5 + x3 – 1 is even, odd, or neither. Justify your ans wer.

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Q: Let f(x) = x^2 −x −2. Using the Intermediate Value Theorem, explain why this function has (at least)…

A: The function is  f(x)=x2-x-2

Q: Find the constant a such that the function is continuous on the entire real line. S 8x², f(x) ax -…

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Q: Apply the intermediate value theorem  evaluate the function f(x)=x^4+8x+3 at each endpoint of the…

A: Given,         The function is fx=x4+8x+3 at each endpoint of the interval -1,0.

Q: (b) Suppose that f(x) = SBx? – 3 if if xS2 (Bx+ 2 %3D Find the value of ß so that f(x) is continuous…

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Q: Suppose that f(x) is a differentiable function such that f(4)-5 and f(4)=4. If g(x)=2xf(x), then…

A: Given f(x) is differentiable function such that f'(4)=5 and f(4)=4. Differentiating with respect to…

Q: Determine a point where f(x) has a non-removable discontinuity.   f(x)=x^4+3x+2/x^2−3x−4

A: GIVEN DATA : Function fx=x4+3x+2x2-3x-4        TO FIND : Non removable discontinuity of the fx Non…

Q: Explain why f(x) satisfies the Intermediate Value Theorem.  f(x)=ex - x3 - 8  Use IVT to show that…

A: Given        f(x)=ex - x3 - 8  To prove that f(x) has a zero on [-2, 1]

Q: Find the constant a such that the function is continuous on the entire real line. (5x2, ах — 3, x 2…

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Q: Q3) Test the continuity of the function f(x): -15x <0 0S X< 2 F(x) = 2х— 1 x+ 3 X = 2

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Q: Show that the function f(x)=x4+6x+4 has exactly one zero in the interval [−1, 0].   Which theorem…

A: Intermediate value theorem (IVT) can be used to determine whether a function​ f(x) has any zeros in…

Q: Use the Intermediate Value Theorem to show that f(x) = x2 + x − 2 has at least two zeros in the…

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Q: 6. Use the definition, determine whether the function g(x) = x = 2. Justify the conclusion. x² -…

A: The solution is given in the next step.

Q: Let f(x) = Vx +7 and g(x) = x +3 a) Find (2) (2) show work. b) Give the domain of the function (-)…

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Q: Suppose that f(x) is a differentiable function such that f(3)= 8 and f'(3)=-2. If g(x)=xf(x), then…

A: Product Rule of differentiation : d(u.v)dx = u.dvdx + dudx.v

Q: If function g(x) =- is discontinuous at x=a, find the points of (x-a) x2 -1 (x+4)(2x-3)´…

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Q: 8. Determine the natural domain of the function Vx–1 f (x)= x²-3x+2 in interval notation.

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Q: Show that the function is discontinuous on R. f(x) = { 1; 0; for x € Q for a Q

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Q: Let f(x)=x^3−9x. Using Rolle's Theorem, what are all the values c in the interval [-3, 3] such that…

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Q: What is f (3) When x²-9 X-3 f(x) = x when XX3? Assume that f(x) is a function that is continuous for…

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Q: Use the Intermediate Value Theorem to show that there is a root of the given equation in the…

A: Intermediate Value Theorem : The theorem states that for a function f continues on [a, b], if f(a)…

Q: Prove that f (x) = x2 takes on the value 0.5 in the interval [0, 1].

A: Here given

Q: Use the intermediate value theorem to show that f has a zero between a and b.

A: We have to use the intermediate value theorem to show that the given function : f(x)=x5+x3+x2+x+1 ,…

Q: Find and sketch the domain of the function f (x, y) = In(9 – x² – 9y²).

A: The given function,  f(x,y)=ln9-x2-9y2

Q: Let f(x) = ax* + bx . If (1,6) is a relative extremum point, then b is?

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Q: 1. Let f(x) = 4 – 4x?. Verify that f(r) satisfies the conditions of Rolle's Theorem on the interval…

A: 1. The Given function is f(x)=4x-4x2 Diff. with respect to 'x' f'(x)=ln(2).22x+1-8x

Q: Show that the function f : R → R given by f(x) = x3 is injective.

A: To show the given function is injective(one-one), first calculate the value of function at two…

Q: (b) Suppose that f(x) = ൜βx2 − 3    if x ≤ 2βx + 2      if x &gt; 2 Find the value of β so that f(x)…

A: For continuity, left hand limit=right hand limit=f(x)

Q: What is f'(x) at x = -1 of the function f(x)=x^-2 + (x^2)^(1/2) *

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Q: Find the constant a such that the function is continuous on the entire real line. 5x², ах — 7, f(x)…

A: For a function to be continuous, the value of the left and right limits at critical point needs to…

Q: Let f(x) = Vx - 1 g(x) = v2- x obtaind )(x), and the domain of (x),

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Q: 13. Use the Intermediate Value Theorem to show that f(x) = 2x° – 3 has a zero in the interval [1,2].…

A: Given:-f(x)=2x3-3 using the intermediate value theoremf(x) is continuous in the interval 1,…

Q: Examine whether or not the function fx' - 2r +1 ifxs 2 S (x)= x'-2r +1 [3x-2 if x>2 Is continuous at…

A: Given function, fx=x3-2x+1,     if x≤23x-2,            if x&gt;2 To find the continuity of the…

Q: Show that a function f(x) = ∑ ∞n=2 (1/x-nπ + 1/nπ) is well defined on [0, π]. That is, prove the…

A: Given function,

Q: (b) Suppose that f(x) =  βx2 − 3            if x ≤ 2 βx + 2             if x > 2 Find the value of β…

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Q: (-x3 – 4x + 20 ,x > 3 2 – 7x , x < 3 Determine whether the function f(x) = is continuous at x = 3.…

A: From the given problem: fx=-x3-4x+20 , x&gt;32-7x , x≤3 As we know that:  If limx→a+fx=limx→a-fx=fa…

Q: , for x 1 Let f(x) be a function defined by f(x)=- Discuss the contuniuty of f and f'.

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Q: Suppose that f(x) is a differentiable function such that f'(4)=6 and f(4)=4. If g(x)=xf(x), then…

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Q: Use the Intermediate Value Theorem to show that f(x) = 2x3 − 3 has a zero in the interval [1, 2]

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Q: Show that f(x)= has a zero over the interval [0,1]. f(x) = x³ x2 – 3x +1 - Your answer

A: Explanation of the answer is as follows

Q: Consider the function f defined by the formula ( ax2 + bx – 3b if x 1. Determine the values of a…

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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.