how to solve this math question? increasing, decreasing?

Question

Question is k(t) = 3t^2/3-t

what is the local minimum and maximum also, what is each function to estimate the intervals on which the function is increasing and decreasing?

Answer 1

how to solve this math question? increasing, decreasing?

Question is k(t) = 3t^2/3-t

what is the local minimum and maximum also, what is each function to estimate the intervals on which the function is increasing and decreasing?

USING Calculus:

Local Min/Local Max:

Set the derivative equal to Zero and solve for the x value,

1. then substitute the x-value in to the original equation

1.Find the derivative:

d/dt[3t^2/3-t] = 2t^-1/3-1

2.Set derivative =0

2t^-1/3-1 = 0

t^-1/3 = 1/2

t^1/3 = 2

t = 8

Local min/max is k(8)

k(8) = 3(8)^2/3 - (8)

= 3(4) - 8

= 4

Local Max @ (8,4)

Increasing and Decreasing Intervals:

1.Find the derivative:

d/dt[3t^2/3-t] = 2t^-1/3-1

2.set derivative equal to 0 and Undefined to determine the zero's and vertical asymptotes:

2t^-1/3-1 = 0

t^-1/3 = 1/2

t^1/3 = 2

t = 8

2t^-1/3-1 = Undefined (aka. dividing by zero)

t cannot equal 0

3.Test a number in the derivative before 0, between 0 and 8, and also after 8 to see if the function is increasing or decreasing on that interval:

d/dx_x=-1 2(-1)^-1/3-1 = -3 therefore decreasing between (-∞,0)

d/dx_x=+1 2(1)^-1/3-1 = +1 therefore increasing between (0,8)

d/dx_x=27 2(27)^-1/3-1 = -1/3 therefore decreasing between (8,∞)