f0(x) ⇒ f(x)increasing/decreasingMath165: Business CalculusRoy M. LowmanSpring 2010, Week5 Lec1Roy M. Lowman f0(x) ⇒ f(x)increasing/decreasingf0: increasing/decreasingCritical Numb ersf(x) is increasing at points where f0> 0f(x) is decreasing at points where f0< 0Critical Numbers (CN, xc) occur where f0(x) = 0 or f0(x)is undefined.Critical Numbers are values of x = xcwhere f(x) can changefrom increasing to decreasing or decreasing to increasing.If f(x) is defined at xcthen the point on the graph (xc, f(xc))is a Critical Point (CP).In some cases, there is no point on the graph at a criticalnumber xcRoy M. Lowman f0(x) ⇒ f(x)increasing/decreasingf0: increasing/decreasingCritical Numb ersf(x) is increasing at points where f0> 0f(x) is decreasing at points where f0< 0Critical Numbers (CN, xc) occur where f0(x) = 0 or f0(x)is undefined.Critical Numbers are values of x = xcwhere f(x) can changefrom increasing to decreasing or decreasing to increasing.If f(x) is defined at xcthen the point on the graph (xc, f(xc))is a Critical Point (CP).In some cases, there is no point on the graph at a criticalnumber xcRoy M. Lowman f0(x) ⇒ f(x)increasing/decreasingf0: increasing/decreasingCritical Numb ersf(x) is increasing at points where f0> 0f(x) is decreasing at points where f0< 0Critical Numbers (CN, xc) occur where f0(x) = 0 or f0(x)is undefined.Critical Numbers are values of x = xcwhere f(x) can changefrom increasing to decreasing or decreasing to increasing.If f(x) is defined at xcthen the point on the graph (xc, f(xc))is a Critical Point (CP).In some cases, there is no point on the graph at a criticalnumber xcRoy M. Lowman f0(x) ⇒ f(x)increasing/decreasingf0: increasing/decreasingCritical Numb ersf(x) is increasing at points where f0> 0f(x) is decreasing at points where f0< 0Critical Numbers (CN, xc) occur where f0(x) = 0 or f0(x)is undefined.Critical Numbers are values of x = xcwhere f(x) can changefrom increasing to decreasing or decreasing to increasing.If f(x) is defined at xcthen the point on the graph (xc, f(xc))is a Critical Point (CP).In some cases, there is no point on the graph at a criticalnumber xcRoy M. Lowman f0(x) ⇒ f(x)increasing/decreasingf0: increasing/decreasingCritical Numb ersf(x) is increasing at points where f0> 0f(x) is decreasing at points where f0< 0Critical Numbers (CN, xc) occur where f0(x) = 0 or f0(x)is undefined.Critical Numbers are values of x = xcwhere f(x) can changefrom increasing to decreasing or decreasing to increasing.If f(x) is defined at xcthen the point on the graph (xc, f(xc))is a Critical Point (CP).In some cases, there is no point on the graph at a criticalnumber xcRoy M. Lowman f0(x) ⇒ f(x)increasing/decreasingf0: increasing/decreasingCritical Numb ersf(x) is increasing at points where f0> 0f(x) is decreasing at points where f0< 0Critical Numbers (CN, xc) occur where f0(x) = 0 or f0(x)is undefined.Critical Numbers are values of x = xcwhere f(x) can changefrom increasing to decreasing or decreasing to increasing.If f(x) is defined at xcthen the point on the graph (xc, f(xc))is a Critical Point (CP).In some cases, there is no point on the graph at a criticalnumber xcRoy M. Lowman f0(x) ⇒ f(x)increasing/decreasingOne typical use for f0Where is f(x) inc/dec?Use f0(x) to determine intervals where f(x) is increasing and wheref(x) is decreasing.1st find all critical numbers to determine boundaries on the graphwhere f(x) can change from increasing to decreasing etc.These boundaries, xc, occur where f0(x) = 0 or f0(x) isundefined.these boundaries are the only places where f(x) can changefrom inc to dec or dec to inc.2nd determine the sign of f0(x) at one test value of x betweeneach boundaryif f0(x) = (+) at this test value then it is increasing here andat all x in the same interval. It can only change at theboundaries given by the critical numbers.if f0(x) = (−) at this test value then it is decreasing here andat all x in the same interval. It can only change at theboundaries given by the critical numbers.the function does not always change what it is doing across acritical number.Roy M. Lowman f0(x) ⇒ f(x)increasing/decreasingOne typical use for f0Where is f(x) inc/dec?Use f0(x) to determine intervals where f(x) is increasing and wheref(x) is decreasing.1st find all critical numbers to determine boundaries on the graphwhere f(x) can change from increasing to decreasing etc.These boundaries, xc, occur where f0(x) = 0 or f0(x) isundefined.these boundaries are the only places where f(x) can changefrom inc to dec or dec to inc.2nd determine the sign of f0(x) at one test value of x betweeneach boundaryif f0(x) = (+) at this test value then it is increasing here andat all x in the same interval. It can only change at theboundaries given by the critical numbers.if f0(x) = (−) at this test value then it is decreasing here andat all x in the same interval. It can only change at theboundaries given by the critical numbers.the function does not always change what it is doing across acritical number.Roy M. Lowman f0(x) ⇒ f(x)increasing/decreasingOne typical use for f0Where is f(x) inc/dec?Use f0(x) to determine intervals where f(x) is increasing and wheref(x) is decreasing.1st find all critical numbers to determine boundaries on the graphwhere f(x) can change from increasing to decreasing etc.These boundaries, xc, occur where f0(x) = 0 or f0(x) isundefined.these boundaries are the only places where f(x) can changefrom inc to dec or dec to inc.2nd determine the sign of f0(x) at one test value of x betweeneach boundaryif f0(x) = (+) at this test value then it is increasing here andat all x in the same interval. It can only change at theboundaries given by the critical numbers.if f0(x) = (−) at this test value then it is decreasing here andat all x in the same interval. It can only change at theboundaries given by the critical numbers.the function does not always change what it is doing across acritical number.Roy M. Lowman f0(x) ⇒ f(x)increasing/decreasingOne typical use for f0Where is f(x) inc/dec?Use f0(x) to determine intervals where f(x) is increasing and wheref(x) is decreasing.1st find all critical numbers to determine boundaries on the graphwhere f(x) can change from increasing to decreasing etc.These boundaries, xc, occur where f0(x) = 0 or f0(x) isundefined.these boundaries are the only places where f(x) can changefrom inc to dec or dec to inc.2nd determine the sign of f0(x) at one test value of x betweeneach boundaryif f0(x) =