The function if x E Q is 0 if x E R \Q continuous at every x E R. 1 f(x) =

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

A:

Q: Suppose f is continuous everywhere, and | f(x) dx = -1 and 3 Find f(2)dz. f(x) dx 2. Find dx. Cannot…

A: As per our guidelines we are supposed to solve only one question i.e. the first question. For the…

Q: at

A: We have to choose the correct option from the given options according to the given information

Q: let a function f(x ) be given by f(x) = log(n+x) log(e+x) prove that the f(x) is decreasing on…

A:

Q: Starting from the function f(x) = e¬alx| -L 1 where

A:

Q: e that f(x) is continuous, and f'(x) is given in the figure below. y y = f'(x) 1 3

A: In this question we will use basic of calculus i.e concept of maximum and minimum. Solution of the…

Q: Suppose f(x) is the function such that for a > 0 we have f(x) = (e" – 1) In(x) and for x < 0 we have…

A:

Q: If f(x) = ∫0 x   (1 - t^2)e^t^2 dt , on what interval is f increasing?

A: To determine the interval on which f is increasing. f(x)=∫0x(1-t2)et2dt

Q: ALL of the following are properties of continuous functions except a. h/b is continuous at x = a…

A: given

Q: For what value of e is the function defined below continuous on (-∞0,00)? C= 9. f(x) = [cx + 1,…

A:

Q: Given the function h (x) = : x+1 Over what interval(s) is h (x) decreasing?

A:

Q: If f(x, y) is a function such that • f(0,0) = 0 • There exists e > 0 such that f(x, y) = 2x when y =…

A:

Q: The following table lists the values of functions f and h, and of their derivatives, f' and h', for…

A:

Q: Зx2 — 1 Q7- Let f(x){ cx + d Vx + 8 if x 1 Determine e and d so that f is continuous everywhere (as…

A:

Q: The function f(x) = x + 1/x is O Decreasing on (0,1) O Increasing on (0,1) O Increasing on (-1, 0) O…

A: Answer is written below

Q: iii. Does there exist a function x e X such that 50) = f + [ e*: e x(s) ds x(1) = 1² for every t e…

A: It is given that X=C0,1 with the following norm, x=maxt∈0,1 xt The map T:X→X defined as follows:…

Q: 2. If F and G are differentiable functions such that F'(x) = G'(x), which of the following is true…

A:

Q: If f(x) continuous for x 2 0, and F(s) is defined by F(s) = | f(x) e *dx, where s > 0 Find F(s) for…

A:

Q: The function f(x) = x + 1/x is %3D O Decreasing on (0,1) O Increasing on (0,1) O Increasing on (-1,…

A: In this question we are given with the function and we have to find the graph of the function is…

Q: Let y = f(x) be an positive increasing function defined for x > 0. The length of the curve described…

A: Given:         Length of curve y=f(x) is between x=0 and x=t is equal to14t4+t2+lnt+4+t2−ln2.

Q: is continuous and g and h are differentiable functions, show that ch(x) d (e f(t)dt f(h(x))h'(x) –…

A: In the given question, the concept of the Fundamental Theorem of Calculus is applied. Fundamental…

Q: If f:R→R is a continuous even function, then | F(x)dx is necessarily equal to -5 O10 O Cannot be…

A: A function is even if f(-x)=f(x)

Q: If a function f is not differentiable at x = a E R, then f is not a differentiable function. True O…

A: The given statement is if a function f is not differentiable at x=a∈ℝ, then f is not a…

Q: Find the best linear approximation to f on th e interval [0, 1].                            f (x)…

A:

Q: If f(t) is a real valued function defined for positive values of t then L[f(t)] is given by a. e*…

A: Using definition of Laplace transform for any real valued function ft, L{ft}=∫0∞e-stf(t)dt   , t≥0

Q: 1 If f(x) is a non-negative continuous function for all x, such that f(x) + + f(x+1/2) = 1,0 ≤ x ≤ -…

A: Given that fx is a non-negative continuous function for all x, such that fx+fx+12 = 1 for 0 ≤ x ≤…

Q: Find F as a function of x and evaluate it at x = 0, x = 1/3, and x = T/2. F(x) = cos e de F(x) =…

A:

Q: If f is defined at x = a, then | f(x)dx is equal to O 1 O a O-a

A:

Q: Consider the function xe* if x < 0 g(x) = x³ – 2x2 + x if 0 < x < c , %3D if x 2 c where c is a…

A: Given,gx = xex                            if x≤0x3 - 2x2 + x          if 0&lt;x&lt;c1x…

Q: Find f'_(0) and f'„(0) for the given function f. (If an answer does not exist, enter DNE.) if x 0…

A: Hey, since there are multiple questions posted, we will answer the first question. If you want any…

Q: 1. Is a function f(x)=√x differentiable at a) X = 1 b) X = 0 c) X = -1

A: Since you have posted multiple questions i can do first question as per our company guidelines…

Q: If fand g are twice differentiable functions such that g (z) = e(-) and g" (z) = h (2) ef-), then h…

A: To find the answer we need to differentiate the function g with respect to x twice using the product…

Q: The function f(x) = x + 1/x is 2p O Decreasing on (0,1) O Increasing on (0,1) O Increasing on (-1,…

A:

Q: 4) Find the value of the constant k so that the function (1 – k, if z<3, f(z) = 1– kr, if 123, is…

A: This question is related to limit and continuity of a function, we will use condition for a function…

Q: Given the function y=f(x), y" is found to be positive over the interval (4,8), then over this…

A:

Q: b) Show that the function sin for æ e (0, 1] { f (x) = for æ = 0 is absolutely continuous on [0, 1].

A: Function is called continuous function if the left and right hand limit values at each and every…

Q: 50 if x E Q 1 if x € R\ Q Let f(x)= Prove that y=f(x) is not continuous at any points a E IR.

A: Take p ∈ Q and let (xn) be a sequence of irrationals converging to p. Then f (p) = 1 but f (xn)→ 0…

Q: The linear approximation L(x) of a differentiable function f(x) near the point x = a is L(x) = f'(a)…

A: Given statement is false, because

Q: Let f(x) = g(x) (x - 1)(x - 2)(x – 3)°(x – 4)³, where g(x) = x + bx² + cx + d. If f(x) is…

A: Use the following formulae, to obtain the required result. duv=udv+vdudu+v=du+dvddxxn=nxn-1

Q: let a function f(x) be given by f(x) = log(r +x) prove that f log(e+x) (x)is decreasing on [0,0]| ]

A: The solution for the above question is as shown below.

Q: Suppose f is a differentiable function with f(7)=1 and f '(1)=- Let y= f(x)·tan x+e* •cos x . Then…

A:

Q: (b) A function f AR is continuous at a point c E A if for any e > 0, unere exists A one 8 >0 but de…

A:

Q: (e) If f(x) is continuous at x = 0, then what is the value of n: if 20 if x=0 f(x)=\ 38

A:

Q: If f(x) is a function satisfy ing f(x+a)+ f(x)%=D 0 for all c+b x e R and constant a such that f(x)…

A: The least positive value of c is 2a.

Q: Let f(x), g(x) be two continuously differentiable functions satisfying the relationships f'(x) =…

A: Introduction :

Q: Find the lengths of the functions of x over the given interval. Approximate if necessary. y=e^x on…

A:

Q: (2) Is the function f(x) : R → R, sin a 1 ifa # πh,k & Z, ifa = Tk, k ε Z f (x) Borel measurable?

A: Consider the given function, fx=1sinxifx≠πk,k∈Z1ifx=πk,k∈Z   Thus, the given function have infinite…

Q: Consider the piecewise-defined function Ax if x 0 h(x) е (a) Give values for A and B so that h(x)…

A: Given Data: Function: hx=Axif x≤0e-1/x+Bif x&gt;0 (a) The condition for the continuous function is,…

Q: Give a formula for f(x) given that f is continuous and 2 x +x* + 2 = f (t) dt t+ 2

A:

Q: If f(x) and g(x) are continuous functions, then S) • g(x)] dx = | fO fra f(x)dx• g(x)dx. %3D

A: To state whether the statement is true or false with explanations:

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.