Let f be the function defined on [-1,2] by f(x)= 3x^(2/3) -2x. What is its max and min values?

Answer 1

when you finding the minimum or maximun of a curve y = f(x), always find the dy/dx, then find the value of x when dy/dx = 0

f(x) = 3x^(2/3) - 2x
dy/dx = (2/3)(3x^2/3 - 1) - 2x
= 2x^-1/3 - 2
so dy/dx = 0
2x^-1/3 - 2 = 0
2x^-1/3 = 2
x^-1/3 = 1
x = +/-1
so when x= +1
f(x) = y = 3(1)^(2/3) - 2(1)
= 3 - 2
= 1
when x = -1
y = 3(-1)^(2/3) - 2(-1)
= -3 + 2
= -1

so the points are (1, 1) and (-1, -1), draw a graw and see which point is the highest and the lowest
therefore the maximun value is (1, 1) and the minimum value is (-1, -1)

Answer 2

If this function is a rational function like accurate = 3x^2 and bottom=(3 -2x) then right here works. This function is undefined on the fee of x that makes the denominator 0: 3 - 2x = 0 --> x = a million.5. There are relative min at x = 0 f(0 ) = 0 and a relative max at x = 3 f(3) = -9

Answer 3

4

Answer 4

f `(x) = 2.x^(-1/3) - 2 = 0 for turning point
2.x^(-1/3) = 2
x^(- 1/3) = 1
x^(1/3) = 1
x³ = 1
x = 1 for turning point
f(1) = 1
(1, 1) is a turning point.
f "(x) = (- 2/3).x^(- 4/3)
f " (1) = (- 2 / 3)
Because f "(1) is - ve, (1,1) is a Maximum turning point
End Points
f(- 1) = 3.(-1) + 2 = - 1
Point (- 1,- 1)
f(2) = 3.2^(2/3) - 4 = 0.76
Point (2, 0.76)
Maximum Turning point (1,1)
Thus in [- 1,2]:-
Minimum value is - 1
Maximum value is 1

Answer 5

To differentiate this function just use the power rule. Even fraction powers obey the power rule. One max or min is found by differentiation, the other is given by the limiting domain.

Answer 6

f(-1) = ?
f(2) = 0.76220315590459842425511691781692
f(0) = 0
f'(x) = 2x^-(1/3) - 2 = 0 for relative min or max
x^-(1/3) = 1
x = 1
f(1) = 1
max = 1
min = 0