1. Find the Speed ($f'$): Take the derivative of the original function.
2. Seek the Stillness: Set $f'(x) = 0$ and solve for $x$.
3. Find the Height ($y$): Plug your $x$ back into the Original $f(x)$.
4. Check for Gaps: Look at your derivative—is there any $x$ that makes it undefined? (e.g., a zero in the denominator).
Find the $x$-coordinate of the Critical Point for each function.
$f(x) = x^2 - 10x + 25$
$f(x) = 3x^2 + 12x - 7$
$f(x) = x^3 - 3x^2 + 1$
In the first problem above ($x^2 - 10x + 25$), find the $y$-value at the critical point. Does the graph touch the x-axis ($y=0$) at this point? Why?
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Using your knowledge of parabolas, decide if the critical point is a Maximum or a Minimum.
$f(x) = -x^2 + 8x - 10$
$f(x) = 5x^2 + 20x + 100$
Find the critical points of $f(x) = ∛{x}$ (which is $x^{1/3}$).
A farmer finds that the number of bushels of wheat ($B$) he can harvest depends on the amount of fertilizer ($x$) he uses:
$B(x) = -2x^2 + 40x + 500$.
Task: Find the exact amount of fertilizer ($x$) that will give the farmer his Maximum Harvest. How many bushels will he get?
Objective: Explain Critical Points to a younger student using their own body.
The Activity:
1. Have them walk forward fast.
2. Tell them to turn and walk backward.
3. Ask: "Can you turn without stopping? Even for a tiny second?" (No).
The Lesson: "Every turn in life has a 'Stop' hidden inside it. In math, we call that stop a 'Critical Point.' It's where God helps us change directions."
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