The Model: The stock price \( P(t) \) (in dollars) at month \( t \) is:
1. According to the AI (or your visual check), between which months was the stock price dropping?
👮 Mathematical Proof (The "Audit"):
Investors don't trust pictures; they trust Calculus. The function is decreasing when \( P'(t) < 0 \).
2. Find the derivative \( P'(t) \):
3. Set up the inequality \( P'(t) < 0 \) and solve for \( t \) (factorize the quadratic):
4. Conclusion: The company lost value between Month _____ and Month _____.
Sometimes markets don't move. Consider the function \( y = x^3 \).
1. Find \( \frac{dy}{dx} \):
2. Is this function always increasing? Calculate \( \frac{dy}{dx} \) when \( x = -2 \) and \( x = 2 \).
3. Critical Thinking: At \( x=0 \), the gradient is 0. Does this mean the function stops increasing? Explain.
The Function: \( y = x^3 - 3x^2 - 9x + 10 \)
Step 1: Locate them.
Find \( \frac{dy}{dx} \) and solve \( \frac{dy}{dx} = 0 \).
Stationary points are at \( x = \) ____ and \( x = \) ____.
Step 2: Classify them (Nature).
Find the Second Derivative \( \frac{d^2y}{dx^2} \).
Test your \( x \) values:
Conclusion:
The Maximum point is at \( (\_\_\_, \_\_\_) \) because \( f''(x) \) was (Positive/Negative).
The Minimum point is at \( (\_\_\_, \_\_\_) \) because \( f''(x) \) was (Positive/Negative).
Your Mission: You want to trick the AI. You will ask it to find stationary points for a function that doesn't have any.
1. The Function: Consider \( y = x^3 + 3x + 5 \).
2. Manual Check: Try to find where \( \frac{dy}{dx} = 0 \).
(Hint: Look at the resulting equation. Can \( 3x^2 + 3 = 0 \) ever be true for real numbers?)
3. AI Verification: Ask the AI: "Find the stationary points of y = x^3 + 3x + 5."
4. Report: What did the AI say? Did it agree with your manual check?