MathematicalAnalysis Suppose f (x) is defned on (a, b) and continuous at c, where a 0, f (c) < 0, and f (c) = 0 separately.) (b) Is the condition that f is continuous at c necessary in (a)? Justify your answer.

Q: If f(x) = g(x) for x # c and f(c) # g(c) then either f or g is not continuous at c.

A: Continuous Function:In mathematics, a continuous function means that a continuous change in the…

Q: Let f(x) be a function which is twice differentiable with f"(x) > 0 on the interval [a - 1, a + 1]…

A: Given that f(x) is a function which is twice differentiable with f"(x)&gt;0 on the interval…

Q: True or False, and give a short reason: (a) If a function is differentiable, then it is continuous.…

A: Given problem:- True or False, and give a short reason:   (a) If a function is differentiable, then…

Q: 6 language of a function f being continous at a point zo € dom(ƒ). Let f(x) = 2x² +1. Using the e- &…

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Q: Exercise 1. Let f(x) be the function 2х — 1 if x 2. - (a) Find the value of c that makes f(x)…

A: Given f(x)=2x-1if x&lt;-1cxif -1≤x≤2x-2if x&gt;2

Q: Let f (x) and g(x) be differentiable functions in (a, continuous at a and b, and g(x) #0 in [a, b].…

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Q: Let f:[-1,1]-R be defined by f(x) =x²sin- if x is not zero and f(0) =0, then .A f is differentiable…

A: When a function is one–one, i.e., when the values of the domain are equal, the value of the range…

Q: Q4) let f be non-negative continuous function of [a, o]& k> 0 is constant. IF t f(t) a holds, then…

A: Given f is a non-negative continuous function on [a,∞) and k&gt;0 is a constant. Suppose,…

Q: Let Show f is defined for all x &gt; 0. Is f continuous on (0,∞)? How aboutdifferentiable?

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Q: (b) Consider the function f : R → R, f(x) = { (i) Prove that f is discontinuous at 0. (ii) Prove…

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Q: Investigate the continuity of the following function f(2)= if z +0 if z =0 (i) Is f(z)…

A: Calculate the limit:

Q: Let f(x) = (Vx + b² - b)/x, b > 0. What is the domain of f? (Select all that apply.) O (-∞, ∞)…

A: f(x)=x+b2-bx The given query is to find the domain of f(x) Also, to define the function such that it…

Q: (c) Give one example of a function g(1, y) that is continuous with respect to r and y, but not…

A: We take an example g(x, y)=(x+y)2x2+y2 . It is continuous w.r.to x  and y but jointly not

Q: Where is f(x)= -5 -6z+5 discontinuous?

A: The given function is fx=x-5x2+6x-5. We have to find the point where the given function is…

Q: If f is differentiable and non-negative on an interval I in R then /3f + 1 is differentiable on I.…

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Q: Let f: IR be a function. (a) Prove that if f is differentiable at some c I for which f(c) ‡ 0, then…

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Q: Show that the function r(0) = sec 0- + 5 has exactly one zero in the interval 0,5 every point over…

A: Given:-               Show that the function r(θ) = sec θ-1θ3+5- has exactly one zero in the…

Q: Show that if f"(x) < 0 for x ∈ [a, b], then 1/2[f(a) + f(b)] < favg < f[(a+b)/2]

A: Consider the given information. f"(x) &lt; 0 for x ∈ [a, b], then 1/2[f(a) + f(b)] &lt; favg &lt;…

Q: 1 1

A: Part d) to check continuity at a point say x=a ,use the definition of continuity…

Q: Let  Show f is defined for all x &gt; 0. Is f continuous on (0,∞)? How aboutdifferentiable?

A: Given: fx=1x-1x+1+1x+2-1x+3+1x+4- . . .. To show:f is defined for all x&gt;0. To check: Is f…

Q: a) Give an example of a function f : [−2, 3] → R which is not continuous at 1 but which is…

A: In both part , we will find the example of the function.

Q: (a) g(a) and f(b) = g(b` %3D

A: Define h(x) as h(x)=f(x)-g(x). As both f(x) and g(x) are differentiable, we get h(x) is also…

Q: Let f be a real-valued continuous and differentiable function. Let function g be defined by g(x) =…

A: We have to find  Which statement about this proof is correct?