How Derivatives Affect the Shape of a Graph 6

Question

Consider the function 𝑓(𝑥)=7(𝑥-2)2/3. For this function there are two important intervals: (-∞,𝐴) and (𝐴,∞) where 𝐴 is a critical number.

𝐴 is   

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 

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

f(x) = 7(x-2)^2/3

f'(x) = (14/3)(x-2)^-1/3

Since f(x) is defined at x = 2 and f'(x) is not defined at x = 2, 2 is a critical value of f(x).

f'(x) is never equal to zero.

The only critical value is x = 2.

When x < 2, f'(x) < 0

When x > 2, f'(x) > 0

So, f is decreasing on (-∞,2) and is increasing on (2, ∞).

f"(x) = -(14/9)(x-2)^-4/3 < 0 for all x except x = 2.

f is concave down on (-∞,2) and on (2, ∞)