f(x) = x 3 − 3x 2 − 9x − 1

Question

f(x) = x 3 − 3x 2 − 9x − 1

• Find interval(s) where f(x) is increasing.

• Find interval(s) where f(x) is decreasing.

• Find where f(x) is concave up

. • Find where f(x) is concave down

Answer 1

The intervals where f(x) is increasing is where the slope of the tangent lines are positive and intervals where f(x) is decreasing are where the slope of the tangent lines are negative. We find the slope of the tangent line by taking the derivative.:

First, find the critical points (where the derivative equals zero or DNE)

f '(x) = 3x² - 6x - 9

3(x² - 2x - 3) = 0

3(x - 3)(x + 1) = 0

x = 3 and x = -1

(there are no places where the derivative DNE)

Put the critical points on a number line and pick values of "x" in each interval on the number line to see if the derivative is positive or negative)

On Interval 1, let's pick x = -2 then f '(-2) = 3(-2)² - 6(-2) - 9 = 15, a positive number, therefore the function is increasing on this interval. On Interval 2, let's pick x = 0 then f '(0) = 3(0)² - 6(0) - 9 = -9, a negative number, therefore the function is decreasing on this interval. On Interval 3, let's pick x = 4 then f '(4) = 3(4)² - 6(4) - 9 = 15, a positive number, therefore the function is increasing on this interval.

So, the function is increasing on (-∞, -1) U (3, ∞) and the function is decreasing on (-1, 3)

The concavity is determined using the second derivative.

f ''(x) = 6x - 6

The concavity changes at a point of inflection. Possible points of inflection are found by setting the setting the 2nd derivative equal to zero.

6x - 6 = 0

6x = 6

x = 1

Place this possible point of inflection on a number line and look at the concavity on each interval. In this case, there are just two intervals: (-∞, 1) and (1, ∞). x = 0 is a good number for the first interval, Plugging in x = 0 into the second derivative: f ''(0) = 6(0) - 6 = -6, which is a negative number so the function is concave down on the interval (-∞, 1).

Trying a number in the interval (1, ∞) - how about x = 2, we see the f ''(2) = 6(2) - 6 = 6 which is positive, so the function is concave up on (1, ∞)

Answer 2

To find when your function is concave up or concave down, you would find the 2nd derivative f''(x) = 0. This would be f''(x) = 6x - 6 = 0. Solving this, you would get x = 1. Plotting f''(x) = 6x - 6 roughly on the x-axis would show a straight line that intersects the x-axis at x = 1. That straight line is below the x-axis (negative/Concave Down) from (-infinity, 1) and above the x-axis (positive/Concave Up) from (1, infinity).

Sorry, I couldn't squeeze all of that into one 5-minute video.