How Derivatives Affect the Shape of a Graph 5

Question

Consider the function 𝑓(𝑥)=2𝑥+5/3𝑥+3. For this function there are two important intervals: (-∞,𝐴) and (𝐴,∞) where the function is not defined at 𝐴.

Find 𝐴   

For each of the following intervals, tell whether 𝑓(𝑥) is increasing or decreasing.

(-∞,𝐴): Select an answer Increasing Decreasing 

(𝐴,∞) Select an answer Increasing Decreasing 

Note that this function has no inflection points, but we can still consider its concavity. For each of the following intervals, tell whether 𝑓(𝑥) is concave up or concave down.

(-∞,𝐴): Select an answer Concave Up Concave Down 

(𝐴,∞) Select an answer Concave Up Concave Down 

Answer 1

Horizontal asymptote at y = 1

Vertical asymptote at x = -1/3

y-intercept at (0, 5)

x-intercept at (-2.5, 0

Concave down, then asymptote, concave up.

Answer 2

First we need to find A which is your critical point. To find a critical point, we take the derivative of the original function and set it = 0 or where it does not exist.

Using the quotient rule, we find the derivative of the original function is:

2(3x+3) - 3(2x+5)/(3x+3)^2 --> This simplifies down to -9/(3x+3)^2

Setting that equal to 0, we see that since -9 is in the only term in the numerator there will will not be a value of x that would cause this function to = 0. Then we look at the denominator, setting it =0 we get that x cannot equal -1 and that there is a vertical asymptote there.

Putting it on a number line with -1 as the CP, testing -2 and 0 we both get negative, so the function is entirely decreasing from (-inf,-1)U(-1,inf)

Taking the second derivative to find concavity, i brought the denominator up to use the chain rule instead or the quotient rule again. So you would rewrite is -9(3x+3)^-2.

f"(x) = -9(-2)(3x+3)^-3 * 3 --> Simplifies to 54/(3x+3)^3

Setting that =0 you would get x cannot equal -1 again. Putting it on a second number line and testing -2 in the second derivative equation gets you a negative number while test 0 gets you a positive number. So the function is concave down from (-inf,-1) and concave up from (-1,inf).

Hope this helps.