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Section 3.1 – Extrema on an Interval

Section 3.1 – Extrema on an Interval . Maximum Popcorn Challenge .

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Section 3.1 – Extrema on an Interval

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  1. Section 3.1 – Extrema on an Interval

  2. 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 varies from box to box Draw a Picture 11 – 2x Eliminate Variable(s) with other Conditions x Use Calculus to Solve the Problem x 8.5 – 2x 8.5 x x The slope of a tangent is 0 at a max 11 What needs to be Optimized? Quad. Form. You can’t cut an 4.9x4.9 in. square out of an 8.5x11 in. paper Volume needs to be maximized: 1.585 in x 7.829 in x 5.329 in

  3. 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?

  4. Extrema of a Function f(c) 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 I I c f(c) 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.

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

  6. Example 2 The function may have a limit at the highest point BUT there is no absolute maximum value The function may have a limit at the lowest point BUT there is no absolute minimum value f(x) 0.5 .9 .9 .1 .1 .99 .99 .01 .01 x .999 .999 .001 .001 1 .9999 .9999 .0001 .0001 … … … … Absolute Maximum: None Absolute Minimum: None The graph of a function f is shown below. Locate the extreme values of f defined on the open interval (0,1).

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

  8. 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.

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

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

  11. 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.

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

  13. 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)

  14. 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. 2 is a critical number and (2,3) is a critical point -3 is a critical number and (-3,7) is a critical point

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

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

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

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

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

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

  21. 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.

  22. 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

  23. White Board Challenge Find the derivative of the function below:

  24. How do we Find Absolute Extrema? Absolute Maximum f(x) On a closed interval, extrema exist at endpoints or at relative extrema. x a c d e f b Absolute Minimum Absolute Maximum: f(b) Absolute Minimum: f(d) 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?

  25. 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]: • Find the values of f at the critical numbers of f in (a,b). • Find the values of f at the endpoints of the interval. • 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.

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

  27. 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 Domain of f: All Reals Find the values of f at the endpoints smallest largest Not a critical point since it’s an enpoint Answer the Question The maximum occurs at x=2 and is 11; the minimum occurs at x=-1 and 1 and is 2 smallest

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

  29. 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 Domain of f: All Reals Find the values of f at the endpoints largest x=0 is a critical number too since it makes the derivative undefined. Answer the Question The maximum occurs at x=-1and is 7; the minimum occurs at x=0and is 0 smallest

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