Calculus Concavity

Question

Let f be the function defined by

f(x) = x^3 - x^2 - x + k

where k is a constant.

On what interval(s) is f increasing ?

1. (-1, 1/3)
2. (-infinity, -1), (-1/3, infinity)
3. (-1/3, 1)
4. none of these
5. (-infinity, -1/3), (1, infinity)

On what interval is the graph of f concave up?

Answer 1

To determine the intervals for which the function is increasing, we need to calculate the first derivative and find where it is positive (f'(x) > 0). The first derivative is essentially the slope of the function (or the slope of the tangent line at a particular point on the curve). Where the slope is positive, the function will be increasing (and likewise, negative slope implies the function is decreasing). Let's calculate the first derivative.

f(x) = x³ - x² - x + k

f'(x) = 3x² - 2x - 1

Now we need to find where the first derivative is positive. We can do this by setting the derivative equal to zero and solving for x. Then we will test points on each "side" of our x intercept values to see if that particular range is positive or negative. It will be easier to demonstrate than explain using words.

0 = 3x² - 2x - 1

0 = (3x+1)(x-1), this equation factors, but you could use the quadratic formula as well

x = -1/3, 1

Now we test one point between -∞ and -1/3, one point between -1/3 and 1, and one point between 1 and ∞ (we test these in the first derivative).

f'(-1) = 3*(-1)² - 2(-1) - 1

f'(-1) = 3 + 2 - 1 = 4, or more importantly positive (we do not really care about the value, only its sign)

f'(0) = 3(0)² - 2(0) - 1 = -1, negative

f'(2) = 3*(2)² - 2(2) - 1

f'(2) = 12 - 4 - 1 = 7, positive

So, the function is increasing from (-∞, -1/3) and (1, ∞).

Concavity is determined very similar to increasing/decreasing, except it uses the second derivative. Positive second derivative is concave up and negative second derivative is concave down.

f'(x) = 3x² - 2x - 1

f''(x) = 6x - 2

0 = 6x - 2

x = 1/3

Test points similar to first derivative:

f''(0) = 6(0) - 2 = -2, negative

f''(1) = 6(1) - 2 = 4, positive

So the function is concave up from (1/3, ∞).