How Derivatives Affect the Shape of a Graph

Question

Consider the function 𝑓(𝑥)=𝑥2𝑒6𝑥.

𝑓(𝑥) has two inflection points at x = C and x = D with C < D

where C is   

and D is   

Finally for each of the following intervals, tell whether 𝑓(𝑥) is concave up or concave down.

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

(𝐶,𝐷): Select an answer Concave Up Concave Down 

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

Answer 1

Remember that inflection points occur where the 2^nd derivative is equal to zero and the concavity changes from positive to negative or vise versa. So first find the second derivative.

f'(x) = e^6x(6x²+2x)

f''(x) = e^6x(36x² + 24x + 2)

So if you solve for where that's equal to zero, you should get your C and D. I'm just using quadratic rule on the quadratic part (e^6x is always positive), and finding that x = (-2 ± √2)/6. Lastly, you'll want to determine whether the derivative is positive (concave up) or negative (concave down) for before, between, or after these points. Noting that it is a parabola opening up, it will be negative between and positive before and after.