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Q: Show that there is a number between -2 and -1 whose double is exactly 1 less than its cube, and use…

A: Let, f(x) = x3-2x -1. Now, the required number must be root of the equation f(x) because x3-1 = 2x…

Q: Consider the graph of y = x² on the interval [-2, 2]. Using 2 rectangles of equal width, give the…

A: Find the lower sum of f(x) with 2 rectangles

Q: Examine the graph shown. Estimate the value c guaranteed by the Mean Value Theorem, and also f'(c),…

A: Given graph of function

Q: Activity 6: Find the equation of the hyperbola with the following inf 1. Center at (1,2), vertices…

A: According to Bartley guidelines here I done only first question. Kindly repost remaining…

Q: Consider the function f(x) =x? on the interval 0. . According to the Mean Value Theorem, there must…

A:

Q: Find a one-to-one correspondence between [0, 1] and (0, 1). Find a one-to-one correspondence between…

A:

Q: Show that there is a correspondence between the two intervals [0, 1) and [0,1].

A: Given interval 0≤x<1 is a subset of  0≤x≤1. In this situation ,there is an obvious injective…

Q: - (10 points) Show |a – b| < |a| + [b| Va,b eR.

A: We have to show that  a-b≤a+b      ∀a, b∈ℝ

Q: The function f is defined by f (x) = –x² – 3 for all real numbers x. Which of the following…

A:

Q: Use the mean value theorem to show that |sin a – sin b| < |a real values of a and b where a + b

A: let f(x)= sin x f is continuous  on [a,b] and differentiable  on (a,b) Therefore there exist  c such…

Q: (7) If x < 0 then 1/x < 0. (8) If 0<x<y then 0 <1/y < 1/x

A:

Q: (ii) For what values of a, m and b does the function x = 0 0 < x < 1 3 f(x) : -x² + 3x + a тх + b…

A:

Q: Let f(x) = x³ - 10, x= [-15√3, 15√3]. Find the value of c guaranteed by the Mean Value Theorem on…

A: The given function fx=x3-10, x∈-153, 153. We have to find the value of c guaranteed by the Mean…

Q: Does the Mean Value Theorem apply to f(x) = |x – 7| on the interval [1, 9]? Explain.

A: No

Q: The value of C obtained when we apply the Mean Value Theorem for f(x) 32x2 +3 on the closed interval…

A: .

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.