Lesson 15: Second Derivative Test and Optimization

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Presentation transcript:

Lesson 15: Second Derivative Test and Optimization MATH 1314 Lesson 15: Second Derivative Test and Optimization

roots(C'(x),20,60) C''(49) C(49)

Popper 17 Consider the graph of 𝑓 𝑥 =10 𝑒 −.2𝑥 .2𝑥 which models the sales an item x months after it has been first marketed. Find all critical values: 0 b. 2.5 c. 5 d. both (a) and (b) e. None of these 2. What is the value of f”(x) at the critical value? f”(x) < 0 b. f”(x) > 0 c. f”(x) = 0 3. Classify the critical value in terms of f(x). Relative Minimum b. Relative Maximum c. Neither

Popper 17 continued: 4. What is the maximum value? 2.5 b. -0.3431 c. 4.2888 d. 0 5. Determine the intervals where f(x) is increasing and decreasing: inc: (0, 2.5); dec: (2.5, ∞) b. inc: (2.5, ∞); dec: (0, 2.5) c. always increasing d. always decreasing 6. Is the extremum we found an absolute extremum? a. Yes b. No

root(A'(x)) A''(32.5) 65-32.5 32.5*32.5 A(32.5)

Popper 18: Two numbers have a difference of 25. Find their minimum product. Assuming the smaller number is x, what is the larger? 25x b. x – 25 c. 25 – x d. x + 25 2. Write a formula for their product. p = x2 + 25x b. p = 2x + 50 c. p = x2 + 25 d. p = 25x 3. Find the derivative of the product function: a. p’ = 2x b. p’ = 2x + 25 c. p’ = 27x d. p’ = 25

Popper 18, continued. 4. Determine the critical values. 23 b. 12.5 c. 0 d. 50 5. Determine the second derivative of the product function: p” = 25x b. p” = 2x c. p” = 2 d. p” = 2x + 25 6. Is the critical value a minimum or maximum? a. It is a minimum b. It is a maximum