Chapter 5 Section 1 I N C R E A S I N G A N D D E C R E A S I N G F U N C T I O N S Increasing and Decreasing Functions A function is increasing if the graph goes up the y value from left to right and decreasing if its graph goes down from left to right Let f be a function defined on some interval Then for any two numbers x1 and x2 in the interval f is increasing on the interval if f x1 f x2 whenever x1 x2 and f is decreasing on the interval if f x1 f x2 whenever x1 x2 Example Test for Intervals Where f x is Increasing and Decreasing Suppose a function f has a derivative at each point in an open interval then xf 0 if for each x in the interval f is increasing on the interval xf 0 if for each x in the interval f is decreasing on the interval 0 xf if for each x in the interval f is constant on the interval Critical Numbers The critical numbers for a function f are those numbers c in the domain of f for which cf or 0 cf doesn t exist A critical point is a point whose x coordinate is the critical number c and whose y coordinate is f c Note Not all critical numbers will be an extreme point mentioned in the next section Applying the Test 1 Locate the critical numbers for f on a number line as well as any points where f is undefined These points determine several open intervals 2 Choose a value of x in each of the intervals determined in Step 1 Use these values to decide whether or in xf xf that interval 0 0 3 Use the test on the previous 2 slides to decide whether f is increasing or decreasing on the interval Chapter 5 Section 2 R E L AT I V E E X T R E M A Relative Maximum or Minimum Let c be a number in the domain of a function f Then f c is a relative maximum for f if there exists an open interval a b containing c such that f x f c for all x in a b Also f c is a relative minimum for f if there exists an open interval a b containing c such that f x f c for all x in a b If c is an endpoint of the domain of f we only consider x in the half open interval that is in the domain Example x1 would be a relative max as well as x3 since it would have the highest y value for all x in the open interval x2 would be a relative min as well as x4 Another Example w disagreement There is disagreement on calling an endpoint x4 a maximum or a minimum However we may define it has a relative min or relative max because this is an applied calculus book which should be considered a max or a min of the function This will be mentioned in Chapter 6 Relative Extremum If a function f has a relative extremum max or min at c then c is a critical number or c is an endpoint of the domain NOTE We can not say the reverse way First Derivative Test Let c be a critical number for a function f Suppose that f is continuous on a b and differentiable on a b except possibly at c and that c is the only critical number for f in a b 1 f c is a relative maximum of f if the derivative is positive in the interval a c and xf negative in the interval c b 2 f c is a relative minimum of f if the derivative is negative in the interval a c and xf positive in the interval c b Relative Extrema cont The last 2 that we saw shows that there is a critical point but the critical point is not a relative extreme point Chapter 5 Section 3 H I G H E R D E R I VAT I V E S C O N C AV I T Y A N D T H E S E C O N D D E R I VAT I V E T E S T Higher Derivatives f f f If a function f has a derivative then the derivative of if it exists is the second derivative of f written The derivative of if it exists is called the third derivative of f and so on By continuing this process we can find fourth derivatives and other higher order derivatives f Notation for Higher Order Derivatives The second derivative of y f x can be written using any of the following notations f x Dor xf 2 x 2 yd 2 dx The third derivative can be written in a similar way For n 4 the nth derivative is written f n x Note is not the same as 4 x 4 x f f Concavity of a Graph The following examples are graphs of increasing functions The only difference between these 2 graphs is its concavity Concavity Second Derivative The second derivative gives the rate of change of the first derivative i e the rate at which the rate of change is changing It indicates how fast the function is increasing or decreasing The rate of change of the derivative affects the shape of the graph cont A function is said to be concave up on an interval a b if the graph of the function lies above its tangent lines at each point of a b A function is said to be concave down on an interval a b if the graph of the function lies below its tangent lines at each point of a b A specific point A point where a graph changes concavity is called an inflection point 2 Examples of Concave Up Functions 2 Examples of Concave Down Functions Test for Concavity f Let f be a function with derivatives of f and existing at all points in an interval x 0 a b Then f is concave up on a b if f for all x in a b and concave down on a b if for all x in a b x 0 f Finding the Concavity of a function at Open intervals The way of finding the concavity of a function at open intervals is done the same way as the first derivative First find the values of c that will make or doesn t exist f 0 x f x f x 0 If for all x in the open interval a b then the function is concave up f x 0 If for all x in the open interval a b then the function is concave down Inflection point At any inflection point for a function f the second derivative is 0 or does not exist An inflection point is found if a …