Decide whether or not the real function In (cos 2x) 2x2 x + 0, f(x) = -1, = 0, x = is continuous at x = 0.

Q: The absolute minimum value of the function f(x) = –3 sinx on the interval [2, 8] is

A: To find the absolute minimum value of f(x), first we have to find the derivative of f(x) i.e. f'(x).…

Q: Explain why the function has at least one zero in the given interval.

A: Given function is  fx=x2-2-cosx over interval 0, π.

Q: Determine if the function f(x)= is continuous -- sin(x) x on the interval I= (0,1).

A:

Q: Find the constant a such that the function sin x -13 - S(x) = %3D a+ 11x, x 20 is continuous on the…

A:

Q: Find the open intervals on which the function is increasing or decreasing.

A: Given: fx=sinx-1, 0<x<2π To Find: Increasing and decreasing function.

Q: Show that the function is continuous. (x) = 3x cos 3x

A: Show that the function is continuous. (x) = 3x cos 3x

Q: Use the Second Fundamental Theorem of Calculus to find F'(x).

A: The function is given as

Q: If f"(x) = V - cos(x). f'(0) = 2, and f(0) = 0, find f(x). %3D

A:

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

A:

Q: B) Show that fx) = sin for x * 0. for x= 0. X = is not continuous.

A:

Q: Define a function f : R² → R² by f 2.у 1. Show that f is differentiable on A = E R² |x > 0, y > 0}…

A: To show that the function f defined as below is differentiable on A and compute its derivative

Q: Determine if the function is continuous at point (0,0): F(x, y)=**y°cos(x*+y³) (x²+y²)2

A: Given that Now take x=0, Now take y=0,

Q: Find the anti-derivative of g(x) = sec(x) tan(x) which contains the point (TT, 0). O G(x) = -csc(x)…

A: Solution to the anti-derivative can be found by integrating the function.

Q: Determine the domain of the function and prove that it is continuous on its domain f(x) = cosx2 /…

A: f(x)=cosx2x2-1

Q: Find the derivative of the function. f(x): f(x) = 2xarccos(x) – 2/1 – x² 2arccos(x) + 2 2arc cos (x)…

A:

Q: Find the ninety-ninth derivative of the function ƒ (x) = cos(2x). O f(99) (x) = 2 sin(2x) 99 f(99)…

A: Given : f(x) = cos (2x)    Times = 99

Q: Show that the function is continuous on  (∞ , -∞).

A:

Q: Determine if the function is continuous at point (0,0): F (x, y) _x³y°cos(x³+y³) (x²+y²)²

A:

Q: Explain why the function ƒ(x) = sin (1/x) has no continuous extension to x = 0

A: Then we can extend the definition of the function at that point to make it continuous there.

Q: Show that the function is continuous. f(x) = cos(x 2)

A:

Q: Show that f(x) = x – cos(x) has only 1 real zero on the interval [0, .

A: given fx=x-cosxx∈0,π2

Q: Find the global maximum and global minimum of the function f(x)=sinx−x in the interval [−π,π].

A:

Q: 10. Show that f (x) = { x² sin(1/x) if x + 0 if x is differentiable, but the derivative is not…

A:

Q: Differentiate the function f (x) = 4 cos 3x - 3 sin 4x. %3D O -4sin 3x -3cos 4x O -12(sin 3x + cos…

A:

Q: Find an analytic function whose real part is e-x(x sin y-y cos y )

A:

Q: Determine the points in the interval [0, 2π) at which the graph of the function f(x) = √2x + 2…

A: We find point of tangent on given curve.

Q: Find ao of the function f(x) 1-Cosx f(x) =

A:

Q: How must f(0) be determined so that the function f(x) = x2-x ニー x0, is continuous x-1 at X= 0?

A: Let f be a real function . Let a be any point in its domain . If limx→afx exist and equal to fa Then…

Q: Find the value of K that make the function f continuous at x=2 -4 スー3x+2 sin(nx) + kx + k if x 2

A:

Q: Use the first derivatives test to find the local exterma of f(x) =x3-12x on R

A:

Q: If the function f(x) = {cos(2πx−a),x<−1x3+1,x≥−1 is continuous, what is the value of a?

A: fx=cos2πx-a,  x<-1        x3+1,    x≥-1 Since Cosine function is continuous for all real numbers,…

Q: Linearize f(x) = 5 cos x about x = pi over 2. f(x) = -5x

A: linearization around x=a is given by :L(x)=f(a)+f'(a)(x-a)

Q: Use the Fundamental Theorem of Calculus to find the derivative of h(x)= (from -4 to sin of…

A: This is a calculus problem

Q: Prove that the function f(x) = cos x is strictly decreasing on the interval [0,1]. %3!

A: 5) Given that, fx=cosx

Q: Find all the x-intercepts of f(x)=cos(x) on the interval [0,2π)

A:

Q: Find the most general anti-derivative of f(x) = 9/x +1+ co x. %3D 9/x- sin x + C 6xVx + x +sin x 6xV…

A: Here I am attaching image so that you understand each and every step.

Q: Use the First Fundamental Theorem of Calculus to find F"(x) if = [*_ sinh√/1 +ť² dt. F(x) =

A:

Q: The function f (x) ,x € [-27, 2n] is continuous on: 1-cos x

A:

Q: Show that x³, if x 0 is not continuous at x = 0. %3D

A: According to the problem, we have

Q: Find the interval in which function f(x) =sin (logex)+cos(logex) is decreasing

A:

Q: Find the ninety-ninth derivative of the function f (x) = cos(2x) . O f(09) (x) = 2% sin(2x) f(99)…

A: Let's find.

Q: defines 2 as y. y- cos (3x2) = 5 a function of x and • differentiate implicity to find az ду

A:

Q: B) Show that f(x) sin for x * 0. for x= 0. is not continuous.

A:

Q: Use implicit differentiation to prove that d/dx [ xn ] = nxn − 1

A: Here we use implict differentiating to prove thatddxxn = nxn-1

Q: Suppose that f(x) is a continuous function on the interval [0, T] . If f(x)=(sin x)ª for 0<x<T, then…

A: we have to find the value of given function. the solution is following.

Q: Let f : (0,∞) → (0,∞) be given by f(x) = ln(2 + x). Prove that f has a fixed point located between 1…

A: The given problem is to prove that the function f(x) has a fixed point in between 1 and 1.2 Given…

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.