Calculus with Algebra and Trigonometry II Lecture 2 Maxima and minima, convexity/concavity, and curve sketching Jan 22, 2015 Calculus with Algebra and Trigonometry II Lecture 2Maxima andjan minima, 22, 2015 convexity/concav 1 / 15
Maxima and minima A function has a global maximum at a point c if f (c) f (x) for all values of x. A function has a global minimum at a point c if f (c) f (x) for all values of x. A function has a local maximum at a point c if there is an open interval (c ɛ, c + ɛ), then f (c) > f (x) for all values of x in the interval i.e. c ɛ < x < c + ɛ. A function has a local minimum at a point c if there is an open interval (c ɛ, c + ɛ), then f (c) < f (x) for all values of x in the interval i.e. c ɛ < x < c + ɛ. Calculus with Algebra and Trigonometry II Lecture 2Maxima andjan minima, 22, 2015 convexity/concav 2 / 15
Critical points The point (c, f (c) is a critical point for the function f (x) if f (c) = 0. At a non critical point the local behaviour of the function is approximated by the tangent line. At a critical point the tangent line is horizontal and you can t which of the three following possibilities describes the behaviour of the function near the critical point Calculus with Algebra and Trigonometry II Lecture 2Maxima andjan minima, 22, 2015 convexity/concav 3 / 15
Convexity and concavity A function is convex if the line joining any two points on the graph lies entirely above the graph A function is concave if the line joining any two points on the graph lies entirely below the graph The convexity/concavity of a function is determined by the second derivative Convex f > 0 Concave f < 0 Calculus with Algebra and Trigonometry II Lecture 2Maxima andjan minima, 22, 2015 convexity/concav 4 / 15
A critical point in an convex interval of a function is a local minimum and a critical point in a concave interval is a local maximum. If the function f (x) has a critical point at x = a then f (a) = 0 and f (a) > 0 local minimum f (a) = 0 and f (a) < 0 local maximum A point where the second derivative is zero is called a point of inflection. It is where the function changes from convex to concave or vice versa. Calculus with Algebra and Trigonometry II Lecture 2Maxima andjan minima, 22, 2015 convexity/concav 5 / 15
Curve sketching To sketch the graph of a function, f (x), use the following checklist Find the zeros of f (x) Determine whether f (x) has any asymptotes-horizontal, vertical, and/or slant Use the first derivative to determine the critical points and intervals where the function is increasing or decreasing Use the second derivative to find the points of inflection and the regions of convexity/concavity Calculus with Algebra and Trigonometry II Lecture 2Maxima andjan minima, 22, 2015 convexity/concav 6 / 15
Asymptotes Horizontal asymptotes A function f (x) has a horizontal asymptote(s) if lim x ± f (x) exists For example f (x) = 2x x 2 + 9 has asymptotes f (x) = 2 and f (x) = 2 since lim x 2x x 2 + 9 = 2 lim x 2x x 2 + 9 = 2 Vertical asymptotes A function f (x) has a vertical asymptote at x = a if lim x a f (x) doesn t exist. Typicall;y rational functions whose denominators vanish at x = a For example f (x) = tan x has a vertical asymptote at x = π 2 since lim x π tan x doesn t exist. 2 Calculus with Algebra and Trigonometry II Lecture 2Maxima andjan minima, 22, 2015 convexity/concav 7 / 15
Slant asymptotes A function f (x) has a slant asymptote if it tends to a staight line as x ±. If the line is y = mx + b then f (x) lim = m lim (f (x) mx) = b x x x infty For example for the function f (x) = 4x 2 + 7x + 18 m = lim x 4x 2 + 7x + 18 x 4x = lim 2 + 7x + 18 x x 2 = 2 b = lim x ( 4x 2 + 7x + 18 2x) Calculus with Algebra and Trigonometry II Lecture 2Maxima andjan minima, 22, 2015 convexity/concav 8 / 15
b = lim x ( 4x 2 + 7x + 18 2x) = lim x = lim x = 7 4 4x 2 + 7x + 18 4x 2 4x 2 + 7x + 18 + 2x 7x + 18 4x 2 + 7x + 18 + 2x So the slant asymptote for f (x) is 4x 2 + 7x + 18 + 2x 4x 2 + 7x + 18 + 2x f (x) 2x + 7 4 x Calculus with Algebra and Trigonometry II Lecture 2Maxima andjan minima, 22, 2015 convexity/concav 9 / 15
Curve sketching example 1 Sketch the graph of the function Zeros: f (1) = 0 thus f (x) factors as f (x) = x 3 3x 2 9x + 11 f (x) = (x 1)(x 2 2x 11) The two other zeros can be found from the quadratic formula to be 1 ± 2 3 Asymptotes Horizontal - No : lim x f (x) Vertical - No : there is no value, a, where lim x a f (x) doesn t exist. Slant - No : lim x f (x) x Calculus with Algebra and Trigonometry II Lecture 2Maxima and Janminima, 22, 2015 convexity/concav 10 / 15
Derivatives f (x) = 3x 2 6x 9 = 3(x 2 2x 3) = 3(x + 1)(x 3) f (x) = 6x 6 = 6(x 1) The crtical points are given by f (x) = 0 x = 1, 3 and the points of inflection by f (x) = 0 x = 1 Calculus with Algebra and Trigonometry II Lecture 2Maxima and Janminima, 22, 2015 convexity/concav 11 / 15
Putting it all together we get a graph You can distinguish the types of critical points either by looking at the behavior of f near the point. Thus 1 is a local maximum since the slope goes from positive to negative as you pass though the point. Similarly 3 is a local maximum since the slope goes from negative to positive. Alternatively we can use the fact that the function is concave at -1 to deduce it has a maximum there and 3 is a local minimum since the function is convex there. Calculus with Algebra and Trigonometry II Lecture 2Maxima and Janminima, 22, 2015 convexity/concav 12 / 15
Curve sketching example 2 Sketch the graph of the function Zeros It doesn t have any zeros Asymptotes f (x) = 1 4 x 2 Horizontal - Yes, the x axis since lim f (x) = 0 x ± Vertical - Yes, the lines x = ±2 since lim f (x) x ±2 doesn t exist Slant - No, if it has a horizontal asymptote it cannot have a slant asymptote Calculus with Algebra and Trigonometry II Lecture 2Maxima and Janminima, 22, 2015 convexity/concav 13 / 15
Derivatives f (x) = 2x (4 x 2 ) 2 f (x) = 8 + 6x 2 (4 x 2 ) 3 There is only one critical point x = 0 and no points of inflection Calculus with Algebra and Trigonometry II Lecture 2Maxima and Janminima, 22, 2015 convexity/concav 14 / 15
The graph then looks like Note the one critical poinrt is a local minimum since the function is convex there. Calculus with Algebra and Trigonometry II Lecture 2Maxima and Janminima, 22, 2015 convexity/concav 15 / 15