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

Q: Find the first and second derivatives of the function. f(x) = In(x + 4) f'(x) = f"(x) I3D

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Q: Determine the derivative of f(x) = 3 x2 In x with respect to x. f '(x) :

A: We differentiate both sides with respect to x and apply the rule d/dx(uv)=ud/dx(v)+vd/dx(u)

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

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Q: 4. Is the function given by G(x) 1 x-1 the interval (0, ∞)? Why or why not? continuous over

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Q: Prove, using the definition of derivative, that if f(x)  = Cps x,then f'(x) =- Sin x

A: Given fx=cos xBy definition of derivativef'x=limh→0 fx+h-fxhfx=cos xThereforef'x=limh→0 cosx+h-cos…

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: Find the hundredth derivative of the function f (x) = cos(2x). O f(100) (x) = -2!00 cos(2x) O f(100)…

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Q: Determine the intervals where the function f (x) = x ln x (x 7 0) is increasing and where it is…

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Q: Find the intervals of monotonicity of the function f(x)=x+sinx.

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Q: Find where the function is continuous, but not differentiable..

A: Given,

Q: The function f(x) = x^2 is decrea sin g on (- ∞,0)  True or False

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Q: Find the open intervals on which the function is increasing or decreasing.     f(x) = sin x − 1, 0…

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Q: What value for a will make sinx { ,X # 0 2x f(x) = %3D 3D ,X = 0 a continuous function?

A: For a function f(x) to be continuous at x = a,limx→a-f(x) = limx→a+f(x) = f(a)

Q: A) Consider the function f(x) whose second derivative is f"(x)=4x+3sin(x). If f(0)=4 and f'(0)=4,…

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Q: Suppose that f(x) is a continuous function on the interval [0, π]. If f(x)=(sin x)² for 0<x<, then…

A: Since the f(x) is continuous function in [0, π], therefore the value of function at x = 0 must equal…

Q: Let f(x) = Cx + 7x if x< 4 x³ - cx if x2 4. For what value of the constant c is the function f…

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

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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: Verify Mean value theorem for the function f(x)=4x3+x+2 over t interval [0,1].

A: Prove hypothesis of mean value theorem Differentiate continuous and value of c

Q: Consider Mean Value theorem for derivative for function f(x) 4(1-2x) in interval [0,1]. TCOS(TX) -

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Q: Consider the function f(x)f(x) whose second derivative is f''(x)=4x+3sin(x)f′′(x)=4x+3sin(x). If…

A: Consider the function f(x) It's second derivative is f''(x)=4x+3sin(x)If f(0)=3 and f'(0)=4   We…

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

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Q: Find the local maximum value of f using both the First and Second Derivative Tests. f(x) = x + V1 -…

A: First calculate critical point at which either local maxima or local minima will occur by putting…

Q: Prove, using the definition of derivative, that if f(x) = cos x then f ' (x) = - sin x

A: we have to prove by using the definition of derivative that if f(x)=cosx then f'(x)=-sinx   as we…

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

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Q: Prove that if the function f : I → R has a bounded derivative on I, then f is uniformly continuous…

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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 all x values in interval [0,2 ·¤] on the graph of f(x) = -3·sin(x)·cos(x) where the tangent…

A: Given,            f(x) = -3 sin(x) cos(x)

Q: Differentiate the following implicit function with respect to x: xy - -- cos (y*) a.

A: Solving the first part: Given:

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

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Q: Find the derivative of f (x) = arcsin (x²). 2x f' (x) = 1-a4 f' (x) V1-x4 2x O f' (x) = V1-a² O f'…

A: Use differentiation of trigonometric function

Q: 4. Show that the function is continuous on (-00, 0). f(2) = {0 x* sin(1/x) if x 01 if x = 0 S

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Q: Apply the first derivative test to indentify all relative extrema. (b) f(x)=x²-2+1 (c) f(x) = cos²…

A: Note: As per our company guidelines we are supposed to answer the first question only. Kindly ask…

Q: consider a continuous function y=f(x) whose first derivative is y'=x(x- 2).

A: We can find the interval as below.

Q: Show that the function xsin f(x)%D 0, When x 0 is continuous but not differenti able at x = 0 When x…

A: Let f be a function defined in the neighbourhood of 'a'. Then the function f is said to be…

Q: Find the derivative of the transcendental function. h(x) = 7e cos x h'(x) = %3D

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Q: Find the value of a which makes the function below continuous on (-∞, ∞). f(t) = [a(x²/2) x < 1…

A: We will use the definition of continuity to solve this question.

Q: Determine the interval(s) on which f(x)=square root(x/x-4) is continuous.

A: Given query is to find the interval where function is continuous.

Q: Does f(x) 1-V1-x² continuous on the interval [-1, 1]?

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

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Q: Is the function F(x) as given is constant, increasing, or decreasing on the interval (0, ∞)?…

A: Fx=∫x2x1t dt⇒Fx=lntx2x⇒Fx=ln2x-lnx⇒Fx=ln2xx          ∵lnx-lny=lnxy⇒Fx=ln2

Q: Determine the derivative of the function with a domain {x |x # 0 and x # + nn, n E R}. sec(x) f(x)…

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Q: Find the derivative off with respect to x of f(x) = In x⁹. The derivative off with respect to x of…

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Q: Use the definition of derivative to compute f '(0). e-1/x if x * 0 if x - 0 f(x) f '(0) -

A: The definition of the derivative is given by: f'(x)=limh→0f(x+h)-f(x)h⇒f'(0)=limh→0f(h)-f(0)h  ...…

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.