Show that f(x) has a removable discontinuity at x = 0 and determine what value for f(0) wou make f(x) continuous at a = 0. Must redefine f(0) =

Q: 4. Is the function given by G(x) 1 x-1 the interval (0, ∞)? Why or why not? continuous over

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Q: Show that f1x2 Vin x is right-continuous at x = 1.

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Q: Use the definition to show that the function f (x) = 1/x^2 is continuous on the interval [c,…

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Q: Show that the function f(x) = K. when xw0 when xois discontinuous at x = 0.

A: Solution

Q: 1. Let f(x)=√x-2 / x-4 . Show that f(x) has a removablediscotuiti t x=4   2. Given f(x)= √(x+1).…

A: Here we have to find the derivative of a function at a point.

Q: Sketch the piecewise function defined below and note any type of discontinuity and specify where…

A: Here we want to sketch the piecewise function and want to decide which kind of discontinuity is…

Q: Is the function given by g(x) = x²-3x continuous at x = 4? Why or why not?

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Q: Give an example of a differentiable function ƒ whose first derivative is zero at some point c even…

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Q: Show that the function f(x)=x|x| is continuous and differentiable at x = 0

A: To show that f(x)=xx is continuous at x=0 To show that limx→0xx=00=0 use piecewise definition for…

Q: How many strings of length 5 can be formed if the alphabet (set of available characters)is {a,b,c}…

A: The given alphabets are {a, b, c}. So, the number of characters is 3. The length of the string is 5.

Q: Which of the following statements is false for a function f that can be differentiated at least…

A: We have to define which statement is wrong

Q: Suppose that f is a continuous function such that | f(x) dz = -12. Find (*") dr.

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Q: 6. Use the definition of the derivative to show that the function f(z) = Z is nowhere…

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Q: Let the function f:R-->R be twice differentiable and suppose that f(x)≤0 and f''(x)≥0 for all x.…

A: It is given that, the function f:R-->R be twice differentiable and suppose that f(x)≤0 and…

Q: (D55D4+7D³ + D² - 8D + 4)y = 0

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Q: Determine if the function f(x)= e*¯ er uniform continuous on the interval I= [1,00 [

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Q: Let f be a twice differentiable function on an open interval (a, b). Which statements regarding the…

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Q: F:(- 2, 2] → R, F(x) = [, (t- 3)(f² + 1) dt

A: Given, Fx = ∫-2x2t - 3t2 + 1 dtFx = ∫-2x2t3 - 3t2 + t - 3 dtFx = t44 - 3t33 + t22 - 3t-2x2Fx = x44 -…

Q: Show that the function xsin f(x)%D 0, When x 0 is continuous but not differenti able at x = 0 When x…

A: Let f be a function defined in the neighbourhood of 'a'. Then the function f is said to be…

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.