First Derivative Test Classification

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📚 Part 1: Understanding the First Derivative Test

1. For a stationary point to be a local maximum, what must be true about f'(x) on either side of the point?

f'(x) changes from positive to negative

f'(x) changes from negative to positive

f'(x) remains positive on both sides

f'(x) remains negative on both sides

2. If f'(x) = x(x - 2)(x + 1), the stationary points occur at:

x = 0, x = 2, x = -1

x = 0, x = -2, x = 1

x = 1, x = 2, x = -1

x = 0 only

3. Which of the following statements about points of inflection are correct? (Select all that apply)

They occur where f'(x) = 0

f'(x) does not change sign around these points

They are neither local maxima nor local minima

The function continues in the same direction on both sides

✏️ Part 2: Applying the First Derivative Test

4. Consider f(x) = x³ - 3x² + 2. Find f'(x) and identify all stationary points.

5. For the function in Question 4, use the first derivative test to classify each stationary point. Show your working by testing values on either side of each point.

6. Given f(x) = 2x³ - 6x + 1, complete the following steps:

a) Find f'(x): f'(x) = _______________

b) Solve f'(x) = 0: x = _______ and x = _______

c) Test x = -1.5 in f'(x): f'(-1.5) = _______ (positive/negative)

d) Test x = 0 in f'(x): f'(0) = _______ (positive/negative)

e) Test x = 1.5 in f'(x): f'(1.5) = _______ (positive/negative)

7. Based on your results from Question 6, classify each stationary point:

At x = -1: The point is a _________________ because f'(x) changes from _______ to _______.

At x = 1: The point is a _________________ because f'(x) changes from _______ to _______.

🔍 Part 3: Real-World Application

8. A company's profit function is given by P(t) = -2t³ + 15t² - 24t + 50, where t is time in months and P is profit in thousands of dollars.

a) Find P'(t) and determine when the rate of profit change is zero.

b) Use the first derivative test to classify these points and explain what they mean for the company's profit.

9. Sketch a sign chart for f'(x) = (x + 2)(x - 1)(x - 3) and use it to classify the stationary points at x = -2, x = 1, and x = 3.

First Derivative Test Classification

First Derivative Test Classification

📚 Part 1: Understanding the First Derivative Test

✏️ Part 2: Applying the First Derivative Test

🔍 Part 3: Real-World Application

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