Katrina is studying the effect of foreign substances on bacteria populations. If she introduces a
beneficial substance, such as food, the bacteria population grows. If she introduces a harmful substance,
such as a poison, the bacteria population decreases. The function
f(x) = x4 - 20x3 + 141x2 - 414x+ 432, where x is the time in days, models the bacteria population rate of growth or decrease over time. Determine when the bacteria's population was
increasing.
b) Describe the bacteria population between days 3 and 6. Explain what positive, negative, and values of 0
mean in this situation.
f(x) increasing when f'(x) > 0
f'(x) = 4x^3 - 60x^2 + 282x - 414
= 2(x-3)(2x^2-24x+69)
The roots are 3,6±√(3/2) = 3.00,4.78,7.22
So, f' > 0 in (3,6-√(3/2) U (6+√(3/2),∞)
To determine when the bacteria population is increasing, we need to find the intervals in which the derivative of the given function, f'(x), is positive.
Step 1: Find the derivative of the function f(x) = x^4 - 20x^3 + 141x^2 - 414x + 432.
f'(x) = 4x^3 - 60x^2 + 282x - 414
Step 2: Set the derivative equal to zero and solve for x.
4x^3 - 60x^2 + 282x - 414 = 0
Step 3: Use calculus techniques such as factoring, synthetic division, or the rational roots test to solve the equation. The solutions for x may or may not be real values.
After finding the solutions for x, plot the x-axis on a number line. The intervals between the solutions are the intervals we need to evaluate to determine whether the bacteria's population is increasing.
For the description of the bacteria population between days 3 and 6 (x = 3 to x = 6):
Step 4: Substitute x-values within the given interval into the original function f(x) = x^4 - 20x^3 + 141x^2 - 414x + 432.
For x = 3:
f(3) = (3)^4 - 20(3)^3 + 141(3)^2 - 414(3) + 432
f(3) = 81 - 180 + 423 - 1242 + 432
f(3) = -486
For x = 6:
f(6) = (6)^4 - 20(6)^3 + 141(6)^2 - 414(6) + 432
f(6) = 1296 - 4320 + 1512 - 2484 + 432
f(6) = -2564
The negative values of -486 and -2564 indicate that the bacteria population is decreasing between days 3 and 6. This means that the harmful substance introduced by Katrina is having a negative impact on the bacteria population.
Positive values of the function indicate that the bacteria population is increasing, and it is likely due to the presence of a beneficial substance such as food.
A value of 0 in this situation would indicate a potential change in the trend of the bacteria population growth or decrease. It could indicate a turning point or transition between increasing and decreasing populations.
To determine when the bacteria's population was increasing, we need to analyze the function f(x) = x^4 - 20x^3 + 141x^2 - 414x + 432.
The bacteria population is increasing when the derivative of the function, f'(x), is greater than 0. Let's find the derivative first:
f'(x) = 4x^3 - 60x^2 + 282x - 414
Next, we need to solve the inequality: f'(x) > 0
To do this, we can factor the derivative:
f'(x) = 2(2x^3 - 30x^2 + 141x - 207)
Now, we can use either the factor theorem or synthetic division to find the roots (values of x for which f'(x) = 0), and then determine the intervals where f'(x) is positive.
After performing the calculations, we find that the roots of f'(x) = 0 are x ≈ -2.65, x ≈ 7.15, and x ≈ 13.54.
To determine when the bacteria population was increasing, we need to find the intervals where f'(x) > 0.
Plotting these values on a number line:
<-----(-∞)---(-2.65)---(7.15)---(13.54)---(+∞)----->
Analyzing each interval:
For x < -2.65: f'(x) < 0, which means the bacteria population is decreasing.
For -2.65 < x < 7.15: f'(x) > 0, which means the bacteria population is increasing.
For 7.15 < x < 13.54: f'(x) > 0, which means the bacteria population is increasing.
For x > 13.54: f'(x) < 0, which means the bacteria population is decreasing.
Therefore, the bacteria population was increasing between approximately -2.65 and 7.15, and also between approximately 7.15 and 13.54.
Regarding the description of the bacteria population between days 3 and 6:
To evaluate this, we substitute the values of x = 3, 4, 5, and 6 into the function f(x) = x^4 - 20x^3 + 141x^2 - 414x + 432.
f(3) ≈ 918 (positive value)
f(4) ≈ 72 (positive value)
f(5) ≈ 32 (positive value)
f(6) ≈ 162 (positive value)
Positive values in this situation indicate that the bacteria population is increasing. Therefore, between days 3 and 6, the bacteria population is increasing.
Negative values would indicate that the bacteria population is decreasing, and a value of zero would represent a point where the population does not change, i.e., the population remains constant.