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Section 3.1 – Extrema on an Interval

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Maximum Popcorn Challenge You wanted to make an open-topped box out of a rectangular sheet of paper 8.5 in. by 11 in. The student must cut congruent squares out of each corner of the sheet and then bend the edges of the sheet upward to form the sides of the box. For what dimensions does the box have the greatest possible volume? x x 11 Draw a Picture What needs to be Optimized? Volume needs to be maximized: Eliminate Variable(s) with other Conditions Use Calculus to Solve the Problem 1.585 in x 7.829 in x 5.329 in x x 8.5 8.5 – 2x 11 – 2x You can’t cut an 4.9x4.9 in. square out of an 8.5x11 in. paper x varies from box to box The slope of a tangent is 0 at a max Quad. Form.

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A Beginning to Optimization Problems One of the principal goals of calculus is to investigate the behavior of various functions. There exists a large class of problems that involve finding a maximum or minimum value of a function, if one exists. These problems are referred to as optimization problems and require an introduction to terminology and techniques. Example of an optimization problem: A manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimensions will produce a box with maximum volume?

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Let f be a function defined on an interval I that contains the number c. Then: These values are also referred to as maximum/minimum, extreme values, or absolute extrema. Extrema of a Function f(c) is an absolute maximum of f on I if f(c) ≥ f(x) for all x in I. f(c) is an absolute minimum of f on I if f(c) ≤ f(x) for all x in I. c c f(c)f(c) f(c)f(c) I I

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Example 1 The graph of a function f is shown below. Locate the extreme values of f defined on the closed interval [a,b]. acdefb f(x) x The highest point occurs at x=b The lowest point occurs at x=d Absolute Maximum:Absolute Minimum: f(b)f(b)f(d)f(d)

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Example 2 The graph of a function f is shown below. Locate the extreme values of f defined on the open interval (0,1). 0.5 1 f(x) x The function may have a limit at the highest point BUT there is no absolute maximum value Absolute Maximum:Absolute Minimum: None xy.9.99.999.9999 …… The function may have a limit at the lowest point BUT there is no absolute minimum value xy.1.01.001.0001 ……

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Example 3 The graph of a function f is shown below. Locate the extreme values of f defined on the closed interval [-1,1]. 1 0.5 f(x) x Absolute Maximum:Absolute Minimum: 2None -0.5 The highest point occurs at x=1 & -1 There is no lowest point because a discontinuity exists at the border There is an issue because this function is not continuous on the closed interval [-1,1]

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White Board Challenge Sketch a graph of the function with the following characteristics: It is defined on the open interval (-7,-1). It is not differentiable at x=-4 It has a maximum of 5 and a minimum of -4.

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The Extreme Value Theorem A function f has an absolute maximum and an absolute minimum on any closed, bounded interval [a,b] where it is continuous. acdefb f(x) x Absolute Maximum Absolute Minimum Absolute Maximum:Absolute Minimum: f(b)f(b)f(d)f(d) Key Word. This function is continuous and defined on the intervals.

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Example 1 In each case, explain why the given function does not contradict the Extreme Value Theorem: 1 12 f(x) x Even though the function has no maximum, it does not contradict the EVT because it is no continuous on [0,2]. Even though the function has no minimum, it does not contradict the EVT because it is not defined on a closed interval. 2 12 g(x) x

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White Board Challenge The function below describes the position a particle is moving in a horizontal straight line. Find the average velocity between t = 2 and 4.

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A function f has a relative maximum (or local maximum) at c if f(c) ≥ f(x) when x is near c. [This means that f(c) ≥ f(x) for all x in some open interval containing c.] A function f has a relative minimum (or local minimum) at c if f(c) ≤ f(x) when x is near c. [This means that f(c) ≤ f(x) for all x in some open interval containing c.] Relative Extrema of a Function acdefb f(x) x Typically relative extrema of continuous functions occur at “peaks” and “valleys.” f(c) is a relative maximum at x=c f(d) is a relative minimum at x=d f(e) is a relative maximum at x=e f(f) is a relative minimum at x=f Endpoints are not relative extrema. Plural = Relative maxima/minima

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Relative Extrema and Derivatives Since relative extrema exist at “peaks” and “valleys,” this suggests that they occur when: The derivative is zero (horizontal tangent) The derivative does not exist (no tangent)

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Critical Numbers and Critical Points Suppose f is defined at c and either f '(c)=0 or f '(c) does not exist. Then the number c is called a critical number of f, and the point (c, f(c)) on the graph of f is called a critical point. -3 is a critical number and (-3,7) is a critical point 2 is a critical number and (2,3) is a critical point

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Example 1 Find the critical numbers for. Domain of Function:All Real Numbers Take the Derivative Now find when the derivative is 0 and/or undefined for x values in the domain. Solve the Derivative for 0 The derivative is defined for all real numbers. Both values are in the domain. When is the derivative undefined?

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Example 2 Find the critical numbers for. Domain of Function:All Real Numbers except 2 Take the Derivative Now find when the derivative is 0 and/or undefined for x values in the domain. Solve the Derivative for 0 The derivative is not defined for x = 2. Both values are in the domain. BUT x = 2 is not in the domain of the function. When is the derivative undefined?

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White Board Challenge Consider the function below: Find the equation of the tangent line to the function at the vertex.

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Example 3 Find the critical numbers for. Domain of Function:All Real Numbers greater than or equal to 0 Take the Derivative Now find when the derivative is 0 and/or undefined for x values in the domain. Solve the Derivative for 0 The derivative is not defined for 0 or negative numbers. and 0 2 is in the domain. When is the derivative undefined? Since 0 is in the domain, it is also a critical point.

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Example 4 Find the critical points for. Domain of Function:All Real Numbers Take the Derivative Now find when the derivative is 0 and/or undefined for x values in the domain. Solve the Derivative for 0 The derivative is defined for all real numbers. Both values are in the domain. When is the derivative undefined? Find the y-value(s)

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Example 5 Find the critical numbers for. Domain of Function:All Real Numbers Take the Derivative Now find when the derivative is 0 and/or undefined for x values in the domain. Solve the Derivative for 0 The derivative is undefined for x=-1. Since -1 is in the domain When is the derivative undefined? The derivative never equals 0.

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Critical Number Theorem If a continuous function has a relative extremum at c, then c must be a critical number of f. NOTE: The converse is not necessarily true. In other words, if c is a critical number of a continuous function f, c is NOT always a relative extremum.

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Important Note Not every critical point is a relative extrema. Take the Derivative Solve the Derivative for 0 Find the y-value(s) is NOT a relative extrema

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White Board Challenge Find the derivative of the function below:

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How do we Find Absolute Extrema? Suppose we are looking for the absolute extrema of a continuous function f on the closed, bounded interval [a,b]. Since the EVT says they must exist, how can we narrow the list of candidates for points where extrema exist? acdefb f(x) x Absolute Maximum Absolute Minimum Absolute Maximum:Absolute Minimum: f(b)f(b)f(d)f(d) On a closed interval, extrema exist at endpoints or at relative extrema.

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Procedure for Finding Absolute Extrema on an Closed Interval To find the absolute maximum and minimum values of a continuous function f on a closed interval [a,b]: 1.Find the values of f at the critical numbers of f in (a,b). 2.Find the values of f at the endpoints of the interval. 3.The largest of the values from Steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value.

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Summary of Procedure Find the absolute maximum and minimum of the function graphed below. The value of the function at the critical number 2 is: -3 Find the values of f at critical numbers The value of the function at the enpoint 0 is: 1 Find the values of f at the endpoints The value of the function at the enpoint 3 is: -2 Find the largest and smallest values from the above work smallest largest 1 is the maximum and -3 is the minimum

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Example 1 Find the absolute extrema of the function defined by the equation on the closed interval [-1,2]. Find the values of f at critical numbers The maximum occurs at x=2 and is 11; the minimum occurs at x=-1 and 1 and is 2 Not a critical point since it’s an enpoint Answer the Question Find the values of f at the endpoints smallest largest smallest Domain of f : All Reals

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Example 2 Find the absolute extrema of the function defined by the equation on the closed interval [0,2 π ]. Find the values of f at critical numbers The maximum occurs at x=5 π/3 and is 6.97; the minimum occurs at x= π/3 and is -0.68 Answer the Question Find the values of f at the endpoints largest smallest Domain of f : All Reals

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Example 3 Find the absolute extrema of the function defined by the equation on the closed interval [-1,2]. Find the values of f at critical numbers The maximum occurs at x=-1 and is 7; the minimum occurs at x= 0 and is 0 Answer the Question Find the values of f at the endpoints largest smallest x=0 is a critical number too since it makes the derivative undefined. Domain of f : All Reals

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