(Please only solve using the given formulas in the second attachment) Thank you (P3.1) f(x) for all x in the domain of f.f(c) is an absolute minimum value of fif f(c) 0 on some interval (a,b), then f...

(Please only solve using the given formulas in the second attachment) Thank you

(P3.1)

0 on some interval (a,b), then fis concave up on (a, b). If f"(x) <0 on="" some="" interval="" (a,="" b),="" then="" f="" is="" concave="" down="" on="" (a,="" b).="" second="" derivative="" test:="" suppose="" c="" is="" a="" critical="" point="" of="" some="" function="" f.="" a.="" if="" f"(c)="">< 0,="" then="" f(c)="" is="" a="" relative="" maximum.="" b.="" if="" f"(c)=""> 0, then f(c) is a relative minimum. c. If f"(c) = 0, then the test is inconclusive. (Could be a maximum, minimum, or saddle). L’HOPTIAL’S RULE AREA APPROXIMATION ƒ(x) takes either the The area under a curve can be approximated using rectangles with either right or left endpoints. You can subdivide an interval [a, b] Iff and g are differentiable near a and lim = x-a g(x) 00 or then b - a for n rectangles by taking segments of length - indeterminate form 00 f(x) lim f'(x) = lim x-a g'(x) f(x)dx = lim f(x*)Ax. X-a g(x) n00 i=1 "/>
Extracted text: EXTREMA AND CRITICAL POINTS c is a critical point of fif either f'(c) = 0 or f'(c) does not exist. f(c) is an absolute maximum value of fiff (c) >f(x) for all x in the domain of f. f(c) is an absolute minimum value of fif f(c) < f(x)="" for="" all="" x="" in="" the="" domain="" of="" f.="" f(c)="" is="" a="" relative="" minimum="" value="" of="" fiff(c)="" 2="" f="" (x)="" for="" all="" x="" near="" c.="" f(c)="" is="" a="" relative="" maximum="" value="" of="" fiff(c)="">< f(x)="" for="" all="" x="" near="" c.="" extreme="" value="" theorem="" fermat’s="" theorem="" if="" fis="" continuous="" on="" a="" closed="" and="" bounded="" interval="" [a,="" b],="" then="" f="" iff(c)="" is="" a="" relative="" maximum="" or="" minimum="" value,="" then="" f'(c)="0" attains="" a="" maximum="" and="" minimum="" on="" the="" interval.="" rolle's="" theorem="" mean="" value="" theorem="" let="" f="" be="" a="" function="" which="" satisfies="" the="" following:="" let="" f="" be="" a="" function="" which="" satisfies="" the="" following:="" 1.="" fis="" continuous="" on="" the="" closed="" interval="" [a,="" b].="" 1.="" fis="" continuous="" on="" the="" closed="" interval="" [a,="" b].="" 2.="" fis="" differentiable="" on="" the="" open="" interval="" (a,="" b).="" 2.="" fis="" differentiable="" on="" the="" open="" interval="" (a,="" b).="" 3.="" f(a)="f" (b).="" then="" there="" is="" some="" c="" in="" the="" interval="" (a,="" b)="" where="" then="" there="" is="" some="" c="" in="" the="" interval="" (a,="" b)="" where="" f'(c)="0." f(a)="" –="" f(b)="" f'(c)="а" —ь="" calculus="" and="" graphs="" if="" f'(x)=""> 0 on some interval (a,b), then f is increasing on (a, b). If f'(x) < 0="" on="" some="" interval="" (a,b),="" then="" f="" is="" decreasing="" on="" (a,="" b).="" first="" derivative="" test:="" suppose="" c="" is="" a="" critical="" point="" of="" some="" function="" f.="" a.="" iff'="" changes="" from="" positive="" to="" negative="" at="" c,="" then="" f(c)="" is="" a="" relative="" maximum.="" b.="" iff'="" changes="" from="" negative="" to="" positive="" at="" c,="" then="" f(c)="" is="" a="" relative="" minimum.="" c.="" if="" the="" sign="" of="" f'="" does="" not="" change="" while="" passing="" c,="" then="" f(c)="" is="" neither="" a="" maximum="" nor="" a="" minimum="" (it="" is="" a="" saddle="" point).="" if="" f"(x)=""> 0 on some interval (a,b), then fis concave up on (a, b). If f"(x) <0 on="" some="" interval="" (a,="" b),="" then="" f="" is="" concave="" down="" on="" (a,="" b).="" second="" derivative="" test:="" suppose="" c="" is="" a="" critical="" point="" of="" some="" function="" f.="" a.="" if="" f"(c)="">< 0,="" then="" f(c)="" is="" a="" relative="" maximum.="" b.="" if="" f"(c)=""> 0, then f(c) is a relative minimum. c. If f"(c) = 0, then the test is inconclusive. (Could be a maximum, minimum, or saddle). L’HOPTIAL’S RULE AREA APPROXIMATION ƒ(x) takes either the The area under a curve can be approximated using rectangles with either right or left endpoints. You can subdivide an interval [a, b] Iff and g are differentiable near a and lim = x-a g(x) 00 or then b - a for n rectangles by taking segments of length - indeterminate form 00 f(x) lim f'(x) = lim x-a g'(x) f(x)dx = lim f(x*)Ax. X-a g(x) n00 i=1


Extracted text: 3. Below is the graph of y = f'(x), which is the derivative for some function f. Using the graph of the derivative of f, determine the following. If any cannot be determined based on the graph alone, explain why. y =f'(x) BE CAREFUL WHEN USING [ ] OR ( ) IN INTERVAL NOTATION. a. The number of critical points of f b. The intervals on which f is increasing 5 5 c. The intervals on which f is concave up d. The x values where f has a local minimum 5