For the function given, do the following: (a) find the critical numbers; (b) det
ID: 2841088 • Letter: F
Question
For the function given, do the following: (a) find the critical numbers; (b) determine the intervals of increase and decrease; (c) use the First Derivative Test to determine weather each critical point corresponds to local maximum, a local minimum, or neither.
F(x) = x^(5/3)-3x^(2/3)
For the function given, do the following: (a) find the critical numbers; (b) determine the intervals of increase and decrease; (c) use the First Derivative Test to determine weather each critical point corresponds to local maximum, a local minimum, or neither. F(x) =( x^2)/(x^2+4)Explanation / Answer
For the function given, do the following: (a) find the critical numbers; (b) determine the intervals of increase and decrease; (c) use the First Derivative Test to determine weather each critical point corresponds to local maximum, a local minimum, or neither.
F(x) = x^(5/3)-3x^(2/3)
F'[X] = [5/3][X^(2/3)] - 2[X^(-1/3)] = 0....FOR OPTIMUM ......
X = 6 / 5 = 1.2 ....... IS THE CRITICAL POINT .....
AT X=0 ...F'[X] IS NOT DEFINED BUT F[X] = 0......
WE FIND THAT FOR X< 0 .......F'[X] >0 AND FOR...1.2> X>0 ...F'[X] IS NEGATIVE .
SO WE HAVE ..
X........- INFINITY TO 0.... ..........0...................0 TO 1.2 ........1.2.........1.2 TO INFINTY
F'[X].... ...POSITIVE........UNDEFINED........NEGATIVE..........0.............POSITIVE
HENCE INTERVALS OF INCREASE ARE ....
(-INFINITY , 0 ) ........AND ...(1.2 , INFINITY )
INTERVALS OF DECREASE ARE .....(0 , 1.2)
LOCAL MAXIMUM IS 0 AT X=0
LOCAL MINIMUM IS ... (1.2)^(5/3) - 3*(1.2)^(2/3) = - 2.03264