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Q: 5. Why does it make graphical sense that the derivative of aconstant is zero? That the derivative of…

A: Derivative of a function is a slope of function on given point, or derivative is a rate of change of…

Q: The derivative of the function f is defined by - f'(x) = (x² + 2) sin (2x − 2). If ƒ (2) = 7, what…

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Q: the derivative of E(q) is-(q*(q^2-1))^-1. what's its global maximum and global minimum. Thank you!

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Q: Please provide me with the solution to this problem. Please show the work and try to dumb it down so…

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Q: a. Find the derivative of f(x) = x2 - 5x at x. That is, find f′(x). b. Find the slope of the tangent…

A: Given: fx=x2-5x To Find: a. f'x=?b. Slope of tangent line of the given function at x=-1 and x=3

Q: find the derivative of f (x) at the designated value of x. f (x) = 1x at x = 1/16

A: Given: fx=1x To find the derivative  Differentiate the given function with respect to x we get,…

Q: find the function with the given derivative whosegraph passes through the point P. ƒ'(x) = 2x - 1,…

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Q: The derivative of the function f is defined by f'(x) = (x² + 2) sin (2x − 2). If ƒ (2) = 7, what is…

A: answer is in next step. please give a like!!

Q: Your job is to estimate this special base. In other words, estimate the value of b for which, at…

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Q: Suppose the derivative of the function y = f(x) is y' = (x- 2) (x + 6). At what points, if any, does…

A: see 2nd step

Q: Suppose that the SECOND DERIVATIVE of a function f is given by f" (x) = (x – 2)² (x² – 4x). Find the…

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Q: Sketch the graph of a function z = f(x, y) whose derivative fx is always negative and whose…

A: Given that z=f(x, y) fx is always negativefy is always positive

Q: 2. Use the definition of the derivative to find f'(x) and f"(x). a) f(x) = x? %3D

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Q: find the derivative of f (x) at the designated value of x. f (x) = x3 at x = 12

A: The derivative of a function can be calculated by differentiating both sides of the function with…

Q: Find the derivative of the function.

A: Given that, f(x)=ln(x-1)   As we know, ddxln(x)=1x Chain Rule Of derivative, If…

Q: Find all possible functions f with the given derivative. f'(x)= 1/x-1, x is greater than 1

A: Obtain the all possible functions as follows.

Q: Let f(x) be a function that is differentiable at x = -2. Given that f'(-2) = 3 and f(-2) = 6, find…

A: Given data: f'(-2) = 3 x1 = -2 f(-2) = y1 y1 = 6

Q: find a linearization at a suitably chosen integer neara at which the given function and its…

A: Given         f(x)=sin-1x , a=π12

Q: Find sixth order derivative of the function f(x)=(x-5)^7.cos(5x) by using Leibnitz theorem.

A: n! = n × (n-1) × (n-2) × (n-3) × (n-4) × ............ × 1

Q: find the derivative of f (x) at the designated value of x. f (x) = x + 11 at x = 0

A: To find the derivative of f(x)=x+11 at x=0 Solution: f(x)=x+11 Differentiating both sides w.r.t x we…

Q: The graph of the derivative of a function is given. y For which x is the (original) function…

A: The graph of the derivative (f') of the graph f is given. Now, the original function f is increasing…

Q: The derivative of the function f is defined by ƒ'(x) = (x² + 5) cos (3x). What is the x-coordinate…

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Q: 2. Sketch the graph of a tangent line of the function f(x) = at x = -1. And then find 1- x the…

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Q: find the derivative of f (x) at the designated value of x. f (x) = 1/x at x = 2/3

A: Given: f(x)=1x To find the derivative at x=23

Q: The first derivative of the function h is given by h' (x) = x³ – 3x2 + x. What are all intervals om…

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Q: Find the x-value for all relative extrema of f(x) using the first derivative test. Specify whether…

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Q: The graph below is of f "(x) for some function f(x) with domain (-∞, ∞) (i.e., the graph of of the…

A: Since you have asked a question with multiple question in a single request, we would be answering…

Q: Find a function with the given derivative whose graph passes through point P f'(x)= 2xx+1-cosx,…

A: To determine the antiderivative satisfying the given initial condition

Q: The graph of a function is shown together with the tangent line at a point P. 10 9. 8. 7 6. 4. 3 2 1…

A: Here from Graph Red line tangent to given function f at point p  so derivative of f is equal to…

Q: the first derivative of a continuous function y=f(x) is y'=x(x-12)^2 .find y'' and then use the…

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Q: The derivative of f is equal to f'(x) = f(x)f(-x). Show that f' is not one-to-one. %3D

A: First we will find 1st derivative using the given relationship.

Q: The second derivative of the function g is given by g' (x) = 0.1x³ – 0.29xª – 0.694x³ + 1.9136x². -…

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Q: Find the x-value for all relative extrema of f(x) using the first derivative test, and specify…

A: f(x)=x3/3 + x2/2 -2x Find the x-value for all relative extrema of f(x) using the first derivative…

Q: Suppose the only information we have about a function fis that f(1)=2 and the graph of its…

A: Ti find the linear approximation.

Q: 7. Consider the function f(x) X - 3 a. Using the definition of the derivative at a point, calculate…

A: Given : f(x) = x / x - 3   (1) f' (2)  (2) tangent line when x = 2 (3) normal line when x = 2

Q: 1. Consider the function f(x) = v2 – x. a) Use the limit definition of a derivative to calculate f…

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Q: Find the first and second derivatives of the function. f(x) = x/(7+x2) Identify the intervals on…

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Q: 7)Determine the points at which the graph of the function has a horizontal tangent line. flx) x2 х-1

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Q: Find a formula for (f-l)(x) given that f is one-to-one and its derivative satisfies the equation…

A: We can solve this using idea of function and definition of inverse of function

Q: Suppose the derivative of the function y=f(x) is y’=(x—1)^2(x—2). At what points, if any, does the…

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Q: The derivative of the function f is defined by f'(x) = (x² + 2) sin (2x − 2) . If ƒ (2) = 7, what is…

A: Given f'x=x2+2 sin2x-2. Also given f(2) = 7.   To Find: The absolute minimum value of the function f…

Q: The function f(x) whose derivative is sin x and whose graph passes through the point(0, 2)​

A: we have to find the function f(x) whose derivative is sinx and whose graph passes through the point…

Q: Find the x value for all relative extremes of f(x) using the first derivative test. And specify…

A: Given,            f(x) =x2 - 15x - 15x + 1

Q: Suppose that the first derivative of y = ƒ(x) is y′ = 6(x + 1)(x - 2)2. At what points, if any, does…

A: To determine if the function has any points of maxima, minima or inflection, use second derivative…

Q: Suppose that the first derivative of y = ƒ(x) is y′ = 6x(x + 1)(x - 2). At what points, if any, does…

A: Here the first derivative of the given function is y’= 6 x (x+1) (x-2). If a function has zero…

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 local maximum and minimum values of f using both the First and Second Derivative Tests. (If…

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Q: 3. Write equation of the tangent line to the graph of the function f(x) = ecosx at one of the values…

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Q: suppose the derivative of f exists, and assume that f(2)=3, f'(2)=4. Let g(x)=x^2f(x) and h(x)=…

A: The objective is to find the equation of tangent lines of g(x) and h(x) at x=2

Q: 7. Use the limit definition of the derivative to find f'(x). Then find the equation of the tangent…

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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.