For the function given, do the following: (a) find the critical numbers; (b) det

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

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