In a classroom, 5/8 of the students are wearing blue shirts, and 1/4 are wearing white shirts. There are 36 students in the clas
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Answer:
Please read the complete answer below!
Step-by-step explanation:
You have the following function:
(1)
a) To find the interval on which f is increasing or decreasing, you first calculate the critical points of f(x).
You calculate the derivative f(x) respect to x:
(2)
Next, you equal the derivative to zero, and then you find the roots of the polynomial by using the quadratic formula:
Then, the critical points are x=-1 and x=3
Next, you calculate df/dx for a values of x to the left and to the right of the critical points x1 and x2. If df/dx < 0 the function is decreasing, if df/dx > 0 the function is increasing.
for x = -1.01
Then, in the interval (-∞,-1), the function is increasing
for x = -0.99
In the interval (-1,3) the function is decreasing
for x = 3.01
In the interval (3,+∞) the function is increasing
b) To find the local minimum and maximum you use the second derivative of the function:
(3)
you evaluate the second derivative for the critical points x1 and x2, if the second derivative is positive, you have a local minimum. If the second derivative is negative, you have a local maximum:
for x1 = -1
x=-1 is a local maximum
for x2 = 3
x=3 is a local minimum
c) upward concavity: (-1,3)
downward concavity: (-∞,-1)U(3,+∞)
The inflection points are calculated with the second derivative equal to zero:
For x = 1 you have an inflection point