1. Find all critical numbers of the function g(x) =x3-3x2 2. Find all critical n

Question

1. Find all critical numbers of the function g(x) =x3-3x2 2. Find all critical numbers of the functionx)-sin2x + cos x 3. Locate the absolute extrema of the function f(x)x-3x on the closed interval -2,21 4. Determine whether Rolle's Theorem can be applied to the function fx) r-4 +2) on the closed interval [-2, 4]. IfRolle's Theorem can be applicd. explain why and find all numbers c in the open interval (-2, 4) such that f(c)-0 Determine whether the Mean Value Theorem can be applied to the function f(x) =x. 3x on the closed interval [-1,1] If so, find all numbers c in the open 5. interval (-1, 1) such thatf (c)-(b)-ia)/(b- a) 6. For the funcuon f(o)--6+15

Explanation / Answer

1)g(x)=x3-3x2

differentiate

g'(x)=3x2-6x

for critical numbers g'(x)=0

3x2-6x=0

3x(x-2)=0

x=0,x-2=0

x=0,x=2

2)f(x)=sin2x +cosx

differentiate

f '(x)=2sinxcosx -sinx

f '(x)=sinx(2cosx -1)

for critical numbers f '(x)=0

sinx(2cosx -1)=0

sinx=0,(2cosx -1)=0

sinx=0,cosx =1/2

x=pi , x=pi/3,5pi/3

x=pi/3,pi,5pi/3 are the critical numbers

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