If f(x) = –3x + sin(47x) for x E [0, 1], find the smallest value of c E (0, 1) which satisfies the mean value theorem.

Q: Suppose that f(x) is a function, whose derivative is f '(x)=(1 + cos x)(e* – 1)(x – 2)²(x – 3). At…

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Q: The function f(x) = 3x³ + cos(3ræ) + 2x , x > 1 is known to be a strictly increasing function on its…

A: Given f(x) = 3x3 + cos(3πx) + 2x,  x≥1   ⇒ f'(x) = 9x2 - 3πsin(3πx) + 2

Q: 17. Find all values of the constants m and b for which the function [sin x, mr + b, 12T is a.…

A: Consider the give function y=sinx,          x<πmx+b,      x≥π Since the function fx is continuous…

Q: e^x=3-2x

A: Given; ex=3-2x,(0,1)

Q: Find the absolute maximum and absolute minimum values of f on the given interval. 8. f(x) = 2³3r² -…

A: In this question we have to find the absolute maximum and absolute minimum.

Q: Is there a sequence of continuously differentiable functions: fn : R –→ R such that fn → x and f. →0…

A: Yes, the sequence is given in step 2.

Q: Consider the functionsƒ(x) = sin x and g(x) =4∏2 x(∏ - x).Compute and compare the average values of…

A: Average value of f(x)=sinx on [0,π] is given by: Average value of f(x) is 2/π.

Q: 3 Given the function: f(x) = (1+ x)³/2 . x - 1 Prove that it is increasing on (0, +00) analytically,…

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Q: 1. Find the value/s of b such that the function f is continuous everywhere. x3 + 2, x 0

A: In the question we have to find value of b.

Q: (a) Find the values of α and β such that the function f(x) = ( α cos x + 2 if x < 0 βe3x + αx2 if x…

A: Consider the given piecewise function

Q: Find the linearization L(x) of the function at a. f(x) = cos(x), a = 77/2 L(x) =

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Q: Find the values of α and β such that the function f(x) = ( α cos x + 2 if x < 0 βe3x + αx2 if x ≥ 0…

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Q: Find the characteristic function for fy(x)=e.

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Q: 23 Find a positive value c, for x, that satisfies the conclusion of the Mean Value Theorem for…

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Q: If f(x) is a continuous function on [3,7] and F(x) is an anti-derivative of f(x) such that F(3)=7…

A: Given :f(x) is continuous function 3,7Fx is anti-derivative of fxF3=7 and F7=3To determine…

Q: The function g(x) = ln(x³ – 1) defined on V2, 0) is not zero unless x 2.

A: Given statement

Q: (B) Verify the Mean Value Theorem for f(x) = ln(4x + 1) on | 0,

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Q: Let f(x) = x ln(1+ x*). For all æ € [-1, 1], it is known that In(1 + x) = (-1)" -xn+1 n +1 n=0 2.…

A: Substitute x3 for x in the given power series of ln1+x and simplify.…

Q: Find values of x, if any at which the given function is not continuous. SU) = x2-25

A: Hare we check whether the given function is continuous or not

Q: Find the maximum and minimum values of the function f(x) = x – sin x on [0, 27). %3D

A: The given function is: f(x)=x-sinx For maximum and minimum value, differentiate the function and…

Q: Suppose that f (x) has a continuous second derivative f"(x), f(2) = 4, f'(2) = –1, and / f(x) dx =…

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Q: Find the critical points of the following function. e* + e f(x) = - x

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Q: If we use Simpson's rule for a function f(x) = X +1 with n=4 , then the approximate of S f(x)dx is

A:

Q: Find the absolute maximum and absolute minimum of f(x) on the given interval f(x) = 2x2 – 4.x +3,…

A: To find the absolute minimum and maximum of a continous function on the closed interval [ a, b ], we…

Q: Find the values of α and β such that the function f(x) = ( α cos x + 2 if x < 0 βe3x + αx2 if x ≥ 0…

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Q: The limit of f(x) = sin(x) as x approaches t is O T/x n/2

A:

Q: 9. Suppose f(x) is continuous and f(x)dx = 10, f(x)dx the equation f(x) = 1 has a solution on the…

A: Divide the interval [-5,5] in to two sub intervals [-5,-1] and [-1,5]

Q: Find the absolute extrema of the function on the closed interval.

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Q: Find numbers a and b, and c, so that f is continuous and differentiable at every point. 4) x- + C,,…

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Q: 2.4 Use the definition of the derivative of complex function to show that f(z) is not dif-…

A: The Complex Derivative Definition: Let f be a function defined on a neighborhood of z0 ∈ ℂ. If…

Q: b) Show that the function f (x) = { * x² sin ! for x E (0, 1] for x = 0 is absolutely continuous on…

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Q: 6- if f(x) is a continuous function at x = 0, then a = 1- cos 4x x0 V16+ Vx -4 3.

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Q: Please see picture

A: Introduction:Here, MX (t) denotes the moment generating function (mgf) of the random variable X;…

Q: If b > 3 and average value of f(x) = 6x – x² on the interval [0,b] is 6, then what is b equal to?

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Q: Consider the function given by V2 + x x² – 4 x<-2 g(x) = -2 <x<0. x + 2 2 sin(7x) 0<x Determine if…

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Q: Find the constant a such that this function g(x) is continuous on the entire number line.

A: Given,

Q: Find the values of α and β such that the function f(x) =αcosx + 2 if x < 0  and f(x) = βe3x + αx2…

A:

Q: Q7: Use the MEAN-VALUE-THEOREM to show that x-1<e* < 2x+1, 0<x<ln 2.

A: Since you have asked multiple question, we will solve the first question for you as per our…

Q: Let k be a fixed positive integer. The nth derivative of 1/(xk − 1) has the form Pn(x)/(xk − 1)n+1…

A: Given: nth derivative of 1xk-1=Pnxxk-1n+1 Find: Pn1

Q: Find f(x) if f(3) = 0 and the tangent line at (x, f(x)) has slope x2 + 5 In (14) f(x) =| 히n 글미(구+ 5)…

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Q: a) Find the values of α and β such that the function f(x) = ( α cos x + 2 if x < 0 βe3x + αx2 if x…

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Q: _4. If f(x) is any polynomial of degree 10, then the gth derivative of f(x) is zero. _5. If f'(0) =…

A: Topic = Derivative As per guidelines ,at  a time only three questions can be answered.

Q: 7. Find all values of x on the graph of f(x)=(x-1)´(2x+1) at which the tangent line is horizontal.…

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Q: 1. Suppose that g(x) > 0 for all r, g(x) has a continuous derivative, g(-2) = 5 8 g(4) = 3 and |…

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Q: The Gamma Function is defined by [(x) = fx-¹e-tdt, positive integer then prove that for na T(n) = (n…

A: The given Gamma function is Γx=∫0∞tx-1e-tdt. Prove that for n a positive integer then Γn=n-1!.   As…

Q: If f is a quadratic function such that f(0) = 9 and f(x) x2(x + 1)3 dx is a ratianal.functian,,find…

A: This question is about finding f'(x)  at x=0  where f(x) is a rational function

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.