f(x)=x^3-3x^2-9x+4 find the intervals on which f is increasing or decreasing b. find the local maximum and minimum values of f. c. find the intervals of concavity and inflection points

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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