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Calculus/critical numbers help
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QUESTION: I am having difficulty with the following problem. Any assistance will be greatly appreciated.
find all critical numbers of the function and classify them
g(t)=t sqrt(5-t)
ANSWER: g(t)=t√(5-t)
g'(t)=√(5-t)-[t/√(5-t)]
g'(t)=0 Gives us critical point of t :
√(5-t)-[t/√(5-t)]=0
√(5-t)=t/√(5-t)
5-t=t
5=2t
t=2½ .
g(2½)=2½*√(2½)=3.95
Alon.
---------- FOLLOW-UP ----------
QUESTION: That was the critical point, but how about all the critical numbers like maximum, minimum or neither.
To know if it's a min or max , we derive g(t) twice :
g''(t)= { √(5-t)-[t/√(5-t)] } '=
-1/[2√(5-t)] - { 1/√(5-t) - 1/(5-t)^(3/2) }
= (-3/2)/√(5-t) + 1/(5-t)^(3/2)
= [ -1.5*(5-t)^3 + 1 ] / [ (5-t)^(3/2) ]
g''(2½) = [ -1.5*(5-2½)^3 + 1 ] / [ (5-2½)^(3/2) ]
= [ -1.5*(2½)^3 + 1 ] / [ (2½)^(3/2) ]
= -24.43 / 3.95 = -6.18
g''(2½)<0 That means that the point (2½,3.95) is minimum . ("if g''>0 then max, & if g''=0
then it's saddle").
Alon.
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Answers by Expert:
Alon Mandes
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Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .
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