Suppose f is continuous on an interval containing a critical pointc and f″(c) = 0. How do you determine whether f has a localextreme value at x = c?

Q: Where is f' = 0 ? at x = What are the values in the closed interval [- 1.2, 1.5] where f' undefined?…

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Q: Let g (x) = f' (x) be a continuous function. If the derivative of the continuous function g (x)…

A: Explained below

Q: If f is continuous but not differentiable at x = c then there is a corner on its graph at (c, f(c)).…

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Q: Is f(x)= Vx + 1 continuous at x == -1

A: Draw the graph and check continuty with help of limit of function.

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: Find the critical points and the interval on which the given function is increasing or decreasing,…

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Q: If f is continuous on [a, b] and "If(x)| dx = 0, what can you conclude about f ?

A: Solution:Given If f is continuous on a,b and ∫abf(x) dx = 0To conclude about f ?

Q: Find the critical points of f.  Assume a is a constant. f(x) = (1/21)x21 - a20x

A: Given f(x)=x2121-a20x

Q: Find the global extrema of f(x)=x2e−x on the interval [−1,1].

A: Given function is

Q: Say whether the following is true or false: If c is a critical point of f , then f has a local…

A: If c is a critical point on f then the point c could represent a local minima, a local maxima or a…

Q: true or false  if f(x) is continous at x=a, then f(x) is differentiable at x=a

A: False.    The given statement is not true in general.

Q: Answer the following questions about the functions whose derivativesare given:a. What are the…

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Q: If f(x) is continuous at x=a , then f(x) is also differentiable at x=a.

A: " If F(x) is continuous at x=a then f(x) is differentiable at x=a." The condition for the function…

Q: Is f(x) = -3 A. Yes B. Does Not Exist continuous at x = 3? C. No D. Cannot be determined

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Q: Answer the following questions about the functions whose derivativesare given:a. What are the…

A: f/x=x-12x+2 a). Critical points are the values of x for which first derivative is zero or not…

Q: Use the given derivative to find all critical points of f, and at each critical point determine…

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Q: If f"(c)<0, where c is a critical value and f(x) is a continuous function over an open interval I…

A: We have to find f(c)

Q: For the following function: a - find all critical numbers b - find the open intervals over which the…

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Q: a) Find the critical value(s) of f(x) = x + 2sin(x) on the interval 0sxs 27

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Q: "For a continuous function f(x), defined on an interval a , b, we have d f(x) dx = f(x), dx a…

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Q: Suppose that  ff"' is continuous and f'(c)  =f"(c) =0 , but f"'(c) &gt;0. Does f have a local…

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Q: Let f(x) be a differentiable function for -3<x<1. Over this interval f(x) has exactly one critical…

A: for maxima or minima, we need to check second derivative of the function.

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: Use the given derivative to find all critical points of f, and at each critical point determine…

A: Consider the given data,f'(x)=4-5xx+93and f is continuous everywhere

Q: If f"(c)<0, where c is a critical value and f(x) is a continuous function over an open interval I…

A: The answer is that f ''(c) tells us the concavity of f(x) at x = c. If f(x) is concave up, i.e., if…

Q: Consider the graph of a function f shown below. How many critical numbers does f have on the…

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Q: Given a continuous function of which is twice differentiable, if f" (c) > 0, which of the following…

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Q: Set f(x) = xª for x ≥ 0 and f(x) = 0 for x < 0. For which values of a is f continuous at 0, and for…

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Q: Assume that f'(c) = 0 and that f'(x) changes from + to - at the critical point x = c. This means…

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Q: Consider : f(x)=e 2 , f'(x) =-xe 2 f"(x) = x'e ? Find for f(x): a) The critical number(s). b) The…

A: Consider   fx=e-x22 f'x=xe-x22  f''x=x2e-x22-e-x22 Find   for  fx a The critical Number sSolution:-…

Q: If f(x) =0 at x=c then f has either a minimum or maximum at x=c. If a differentiable function f has…

A: To Determine: true or false                           1) if f'(x)=0 at x=c then f has either a…

Q: 5. Ifc is a critical point of F(x), when the is a local maximum or minimum at c? Explain

A: To give the condition of local maximum or minimum at c if c is a critical point of F(x).

Q: Let f (x) = dt . Find the critical points of f (x).

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Q: Sketch the graph of a function f that is continuous from x = 1 to x = 5 and has the given…

A: To Sketch: The graph of a function f that is continuous from x = 1 to x = 5 and has the given…

Q: Is g(x) = ln( x2+2 ) uniformly continuous on the interval (−∞,∞)?

A: given function is                                                        g(x)=lnx2+2 log is defined…

Q: 3. Suppose that {f(t)} = F(s) with F a differentiable function and that L{f'(t)} Exists. Then…

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Q: If f '(x) > on an open interval extending left from c andf (x) < on an open interval extending right…

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Q: Answer the following questions about the functions whose derivativesare given:a. What are the…

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Q: Find the exact location of all the relative and absolute extrema of the given function. f(x) = x -…

A: Given : f(x)=x-lnxdomain: (0,∞) To find out absolute minimum or maximum , we take derivative with…

Q: Answer the following questions about the functions whose derivativesare given:a. What are the…

A: Given: f'x=x-1e-x we know that critical points are define as points on graph of a function where f'x…

Q: 10 If f is a continuous function and F(x)=f(x) for all x€R, then | f(Gx)dx=?

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Q: Find the critical points of the following function. Assume a is a nonzero constant.

A: Given: fx=xx-a To determine: The critical points of the given function.

Q: Let f(x) = (2 – x)° f(x) has one critical value at A = | 0

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Q: Consider f(x) = xlnx on the interval [1,6]. When will the first derivative equal zero? For each…

A: Given that: f(x) = xlnx   x∈[1,6] To find: When will the first derivative  equal to zero. And for…

Q: . Is f(x)= Vx + 1 continuous at x == -1? Yes, because all functions are continuous No, because the…

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Q: Find the values of a and b such that f is differentiable at x = 3

A: Given,            The function is fx=ax+bx≤32x1-1x&gt;3To find : Find the values of a and b such…

Q: Find all critical numbers for the function f(x) = zt_ z궁 || -

A: The function is given by fx = x13-x-23 To evaluate: The critical numbers of the function.

Q: Answer the following questions about the function whose derivatives are given: (a). What are the…

A: Now, if f'c=0, then c is said to be a critical point of the function fx A function is: Increasing…

Q: Determine unknown constants a and b for which g(x) is continuous at x=1. [5x - 2 x 1

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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.