Answer:
a)
x = -1/√3 = -0.58 is a local maximum
x = 1/√3 = 0.58 is a local minimum
b)
(-1/√3 , 1/√3 ) DECREASING
(-∞, -1/√3) U (1/√3, ∞) INCREASING
Step-by-step explanation:
To answer all of the questions we must obtain the derivative of the fucntion:
If
U(x) = 4(x^3 - x)
then
U'(x) = 4(3x^2 - 1)
U''(x) = 4(3*2 x) = 24 x
U'''(x) = 24
The local maxima and minima of the function U(x) can be found when U'(x) = 0
this occurs when :
3x^2 = 1
that is:
x = ±1/√3
We will know if they are a minimum or a maximum evaluating this points on the second derivative (you can look for it as Second derivative test), if the result is positive the point corresponds to a minimum and if it is negative it will be a maximum.
Here it is easy to determine wheather is a maximum or a minimum because the second derivative is 24x, therefore if x is negative or positive so the second derivative will be.
a)
x = -1/√3 = -0.58 is a local maximum
x = 1/√3 = 0.58 is a local minimum
b)
the intervals at which the function is increasing { decreasing } is given when the first derivative is positive { negative }
The first derivative will be positive when:
3x^2 > 1
|x| > 1/√3 --> x > 1/√3 and x < -1/√3
The first derivatice will be negative when:
3x^2 < 1
|x| < 1/√3 --> x < 1/√3 and x > -1/√3
Therefore the intervals are:
(-1/√3 , 1/√3 ) DECREASING
(-∞, -1/√3) U (1/√3, ∞) INCREASING
** The attached image is a plot of the fucntion where you can see part of the intervals and the local maximum and minimum
