Question about curve sketching with first and second derivatives

[Ginny]

Hi everyone,

I'll post the problem below and then my solution to it. I don't understand, though, why it's wrong...

"A drug that stimulates reproduction is introduced into a colony of bacteria. After t minutes, the number of bacteria is given approximately by N(T) = 1000 + 30 t squared - t cubed. (0 is less than or equal to t is less than or equal to 20).
When is the rate of growth, N'(t), increasing? When is the rate of growth, N'(t), decreasing?"

I found the first derivative N'(t) = 60 t - 3 t squared. I found partition points from that to be 0 and 20.
I made the sign chart:
(- infinity, 0): I tested -1 and found it negative, so decreasing
(0,20): I tested 5 and found it positive, so increasing
(20, infinity): I tested 25 and found it negative, so decreasing

This didn't match the answer in the book (though I thought that only the first deriviative was necessary to determine whether a function was increasing or decreasing -- my first mistake??) so I thought maybe I should include the inflection point from the second derivative. I found the second derivative to be 60 - 6 t, and the inflection point of t = 10.

So then I made another sign chart to include all values of 0, 10 and 20. I evaluated with N'(t) = 60-3 t squared...

(- infinity, 0): I tested -1 and found it negative, so decreasing
(0, 10): I tested 5 and found it positive, so increasing
(10, 20): I tested 15 and found it positive, so increasing
(20, infinity): I tested 25 and found it negative, so decreasing

Based on this, my answer was that the N(t) was increasing over the whole domain of 0 through 20.
But the answer book had "Increasing on (0,10) and decreasing on (10, 20)".
I don't understand why it's decreasing on (10, 20) when I found it to be increasing.
I really appreciate your help and look forward to your answers! Best wishes to you all.

 

[MathLover1]

you should include the inflection point from the second derivative
N''(t) = 60- 6t
has the inflection point of t = 10

Increasing: -infinity <t<10,
Decreasing:10<t<infinity

since it was given a restriction that 0 <= t <=20
you will have:
Increasing: 0 <t<10, or interval (0,10)
Decreasing:10<t<20 or interval (10,20)

 

[Ginny]

Thank you so much for writing to explain, Math Lover. I don't understand, though, why the second derivative equation was used to evaluate whether a function was increasing or decreasing -- I thought that was exclusively what the first derivative was used for, and the second derivative was used to determine whether a function was concave up or concave down.
Is it always the case that when there's an inflection point that the second derivative is used to determine if a function is increasing or decreasing? Again, thank you!

 

[MathLover1]

Inflection points are where the function changes concavity. Since concave up corresponds to a positive second derivative and concave down corresponds to a negative second derivative, then when the function changes from concave up to concave down (or vise versa) the second derivative must equal zero at that point.
So, the point where the function concave up to concave down, shows us where a function is increasing or decreasing .

 

[Ginny]

Thank you so much, MathLover1, this really helped. After I read your explanation, I made a graph of all three (N(t), N'(t), and N"(t)) so I could better understand the important concepts you mentioned. Many thanks to you!