Question

If a number is drawn at random from the set \( \{1, 3, 5, 7, \dots, 59\} \), then the probability that it lies in the interval in which the function \( f(x) = x^3 - 16x^2 + 20x - 5 \) is strictly decreasing is:

• A. \( \frac{1}{5} \)
• B. \( \frac{1}{3} \)
• C. \( \frac{1}{2} \)
• D. \( \frac{1}{6} \)

Hint:

To determine the intervals where a function is strictly decreasing, find the derivative and solve the inequality \( f'(x)<0 \). Then, count the number of favorable outcomes in the given set and calculate the probability.

Answer 1

The Correct Option is D

We are given the function \( f(x) = x^3 - 16x^2 + 20x - 5 \), and we are asked to find the probability that a number drawn at random from the set \( \{1, 3, 5, 7, \dots, 59\} \) lies in the interval where the function is strictly decreasing. 
Step 1: The function \( f(x) \) is strictly decreasing where its derivative is negative. We first find the derivative of \( f(x) \): \[ f'(x) = 3x^2 - 32x + 20 \] Step 2: To find the intervals where the function is strictly decreasing, we solve the inequality \( f'(x)<0 \). First, solve the equation \( f'(x) = 0 \) to find the critical points: \[ 3x^2 - 32x + 20 = 0 \] Using the quadratic formula: \[ x = \frac{-(-32) \pm \sqrt{(-32)^2 - 4(3)(20)}}{2(3)} = \frac{32 \pm \sqrt{1024 - 240}}{6} = \frac{32 \pm \sqrt{784}}{6} = \frac{32 \pm 28}{6} \] Thus, the critical points are: \[ x = \frac{32 + 28}{6} = 10 \quad {and} \quad x = \frac{32 - 28}{6} = \frac{4}{6} = \frac{2}{3} \] Step 3: Now, to determine the sign of \( f'(x) \), we test the intervals formed by the critical points: \( (-\infty, \frac{2}{3}) \), \( (\frac{2}{3}, 10) \), and \( (10, \infty) \).
- For \( x \in (\frac{2}{3}, 10) \), the function is strictly decreasing, as \( f'(x)<0 \).
Step 4: We are interested in the set \( \{1, 3, 5, 7, \dots, 59\} \). The numbers in this set correspond to the odd integers between 1 and 59. These numbers are: \[ 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59 \] Thus, the total number of elements in the set is 30. 
Step 5: The function is strictly decreasing in the interval \( (1, 10) \), which corresponds to the odd numbers \( 3, 5, 7, 9 \). Thus, there are 4 numbers in the set where the function is strictly decreasing.
Step 6: The probability that a randomly selected number from the set lies in the interval where the function is strictly decreasing is the ratio of favorable outcomes to total outcomes: \[ P(strictly decreasing) = \frac{4}{30} = \frac{2}{15} \] Thus, the correct answer is \( \frac{1}{6} \).