2. Which of the following functions satisfy the hypothesis of Rolle's Theorem on the interval [0, 2]? 1 (a) f(x) = |x-1| (b) f(x) = |x – 1| (c) f(x) = x² – 2x (d) f(x) = 2x + 3 2*+3 %3D n→-0 3x-1 3. lim (а) — оо (b) +00

Q: Determine whether the function x2 + 2 x > 2 f(x) = x = 2 is continuous at the 4х — 2 x < 2 given…

A: First of all we will find the left hand limit at X equal to 2 and then find the right hand limit X…

Q: (i) Let f be the function defined on (-∞, ∞) by f(1) = r}(3 – 2)}. (2 – x) -2 Given that f'(x) and…

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Q: If f is continuous and ƒ(5) = 2 and ƒ(4) = 3 then lim ƒ(4x² – 11) = x→2 True False

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Q: Given the function f(x) = x² – 3x f(2+h)-f(2) h (a) Find the expression (b) Use your result from…

A: The given function is ; f(x)=x2-3x.

Q: 11. Find the absolute maximum and minimum values of each function on the given interval: (a) y =…

A: The given functions

Q: 16. Let f(x) be a differentiable function defined on [0, 2] such that f (x) = f (2 - x) for all x e…

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Q: A function f(1) is given by f(1) = h + ab' and has the properties: • f(t) has a horizontal asymptote…

A: NOTE: Refresh your page if you can't see any equations. . the exponential function is given by here…

Q: f(x) = In (x² + 1) %3D Note: 2x and f"(x) = 2(1 – x)(1+ x) (x² + 1)² f'(x) %3D %3D x2 + 1 (a) Find…

A: Here, we are going to solve the first two subparts and for the remaining subpart please post the…

Q: Let f(x) = √38 - x The slope of the tangent line to the graph of f(x) at the point (2,6) is The…

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Q: The slope of the tangent line to the graph of the function y=3V-4at (16,8) is 16.00 4. -1.00 O-0.50…

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Q: 5. If f(x)=3*-9 and g(x)= tanx, then what is the domain of (g+f)(x) on the interval [0,2m]?

A: For the given question, we can write the function (f÷g)(x) as:f(x)g(x)=tanx3x-32Hence, the domain of…

Q: 1. Consider the periodic function f(x) that is defined over the interval (-x, 1) as f(-n < x < x) =…

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Q: 24. If the second derivative f" exists at a value xo, show that f(xo + h) - 2f(xo) + f(xo − h) h²…

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Q: (A) f(x) = √√√x + 2x (B) f(x) = √√√x – 2x² (C) f(x) = 2x² + √x (D) f(x) = 2x − √√x

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Q: f(x.y)= are ) = arccos( x² +y² – 3 )+In(x²+y² –3) fonksiyonunun tanım kümesi aşağıdakilerden…

A: Given function is fx,y=arccosx2+y2-3+lnx2+y2-3. We know that the range of the cosine function is…

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Q: 1. (a) (b) Find and sketch the domain of the function f(x, y) = = Show that x²y² lim (x,y) →(0,0)…

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A: The value of the function at -1 and 1 is 1 and at 2 it is 0.now, the function reaches infinity as x…

Q: B. Calculate the limits of the following functions.. 1. limx¬-1(2X2 + X + 4) 2. limy--2(y3 + 5y – 1)…

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A: Here is the solution of the given problem.

Q: A function f is defined, for x<, by f(x) = 2x² – 6x + 5. (i) Express f(x) in the form a(x – b)² +c,…

A: Please see the below picture for detailed solution.

Q: 7. We define f: R R as follows: {. f(x) = X -x3 if x ≥ 1, if 0 < x < 1, if x < 0. (a) Calculate…

A: Given a function f:R→R defined as follows fx=x3       if x≥1x        if 0≤x&lt;1−x3    if x&lt;0 a…

Q: 3. Consider the function f (x) = 2 x + 1 on the interval [2, 6]. (a) Compute the left-endpoint…

A: Solve the following

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A: Step 15(a) The given functions are:f:R→R:f(x)=x2g:R→R:g(x)=sinxh:R→R:h(x)=3x​To find (h⋅(g⋅f))(x)…

Q: 2. Let 1 f(x): Recall: L(x) = f(a) + f'(a)(x - a) and L(x) f(x) when x is close to a (a) Find the…

A: Given function is -  f(x) = 1/x

Q: Suoppose that f is a differentiable function satisfying the following differential equation: (*) f"…

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Q: (b) Let f:(R\{1}) → (R\{1}) and ƒ˜¹:(R\{1} ) → (R \{1}) be both defined by f(x) = f¯¹(x) = = x+1…

A: As both function and its inverse are given in the question so we have to determine that…

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A: Authoring guidelines: As per the authoring guidelines we can solve only the first three parts of…

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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.