# 📊 The First Derivative Test ## What is the First Derivative Test? The **First Derivative Test** uses the sign of $f'(x)$ to determine whether a critical point is a local maximum, local minimum, or neither. > **💡 Key Idea**: If the derivative changes from positive to negative at a critical point, you have a local max. If it changes from negative to positive, you have a local min! --- ## Understanding the Signs of the Derivative ### What $f'(x)$ Tells Us - **$f'(x) > 0$**: Function is **increasing** (going up) - **$f'(x) < 0$**: Function is **decreasing** (going down) - **$f'(x) = 0$**: Function has a **horizontal tangent** Think of it like driving a car: - $f'(x) > 0$: driving uphill ⬆️ - $f'(x) < 0$: driving downhill ⬇️ - $f'(x) = 0$: at the top or bottom of a hill --- ## The First Derivative Test (Formal Statement) Let $c$ be a critical point of $f$ (so $f'(c) = 0$ or $f'(c)$ undefined). **Test the sign of $f'(x)$ on intervals around $c$**: ### Case 1: Local Maximum If $f'$ changes from **positive to negative** at $c$: - $f'(x) > 0$ for $x < c$ (increasing before) - $f'(x) < 0$ for $x > c$ (decreasing after) - Then $f(c)$ is a **local maximum** 📈➡️📉 ### Case 2: Local Minimum If $f'$ changes from **negative to positive** at $c$: - $f'(x) < 0$ for $x < c$ (decreasing before) - $f'(x) > 0$ for $x > c$ (increasing after) - Then $f(c)$ is a **local minimum** 📉➡️📈 ### Case 3: Neither If $f'$ does **not change sign** at $c$: - Then $f(c)$ is **neither** a max nor a min - Could be an inflection point --- ## Step-by-Step Process ### How to Apply the First Derivative Test **Step 1**: Find all critical points (solve $f'(x) = 0$ and find where $f'(x)$ is undefined) **Step 2**: Create a **sign chart** for $f'(x)$ - Mark critical points on a number line - Test the sign of $f'(x)$ in each interval **Step 3**: Analyze the sign changes - Positive → Negative = Local MAX - Negative → Positive = Local MIN - No change = Neither **Step 4**: State your conclusions --- ## Example 1: Complete Analysis Classify the critical points of $f(x) = x^3 - 3x^2 - 9x + 5$ **Step 1: Find critical points** $f'(x) = 3x^2 - 6x - 9 = 3(x^2 - 2x - 3) = 3(x-3)(x+1)$ Critical points: $x = -1$ and $x = 3$ --- **Step 2: Create sign chart** Test intervals: $(-\infty, -1)$, $(-1, 3)$, $(3, \infty)$ **Interval $(-\infty, -1)$**: Test $x = -2$ $f'(-2) = 3(-2-3)(-2+1) = 3(-5)(-1) = 15 > 0$ ✓ POSITIVE **Interval $(-1, 3)$**: Test $x = 0$ $f'(0) = 3(0-3)(0+1) = 3(-3)(1) = -9 < 0$ ✓ NEGATIVE **Interval $(3, \infty)$**: Test $x = 4$ $f'(4) = 3(4-3)(4+1) = 3(1)(5) = 15 > 0$ ✓ POSITIVE --- **Step 3: Sign chart** ``` -1 3 ++++ | ---- | ++++ ▼ ▲ ``` --- **Step 4: Conclusions** At $x = -1$: $f'$ goes from + to − → **LOCAL MAXIMUM** At $x = 3$: $f'$ goes from − to + → **LOCAL MINIMUM** Calculate the values: - $f(-1) = (-1)^3 - 3(-1)^2 - 9(-1) + 5 = -1 - 3 + 9 + 5 = 10$ - $f(3) = 3^3 - 3(3)^2 - 9(3) + 5 = 27 - 27 - 27 + 5 = -22$ **Answer**: - Local maximum of $10$ at $x = -1$ - Local minimum of $-22$ at $x = 3$ --- ## Creating Sign Charts ### Method 1: Test Points Pick any value in each interval and substitute into $f'(x)$. **Tip**: Use easy numbers like $0, 1, -1, 10, -10$ ### Method 2: Factored Form If $f'(x) = a(x - r_1)(x - r_2)\cdots$, analyze each factor's sign. **Example**: $f'(x) = (x+2)(x-1)$ ``` Interval (x+2) (x-1) Product (-∞, -2) − − + (-2, 1) + − − (1, ∞) + + + ``` --- ## Increasing and Decreasing Intervals The First Derivative Test also tells us where $f$ is increasing or decreasing: ### From the Sign Chart - Where $f'(x) > 0$: $f$ is **increasing** - Where $f'(x) < 0$: $f$ is **decreasing** **Example from above**: - $f$ is increasing on $(-\infty, -1)$ and $(3, \infty)$ - $f$ is decreasing on $(-1, 3)$ --- ## Special Cases ### Case 1: Inflection Point with Horizontal Tangent If $f'(c) = 0$ but the sign doesn't change, it's usually an inflection point. **Example**: $f(x) = x^3$ at $x = 0$ - $f'(x) = 3x^2$, so $f'(0) = 0$ - $f'(x) > 0$ for all $x \neq 0$ (no sign change!) - $x = 0$ is an inflection point, not an extremum ### Case 2: Multiple Critical Points in a Row Handle each one separately using the test. ### Case 3: Derivative Undefined The test still works! Just check the sign change around the point. **Example**: $f(x) = |x|$ at $x = 0$ - $f'(x) < 0$ for $x < 0$ - $f'(x) > 0$ for $x > 0$ - Changes from − to + → local minimum at $x = 0$ ✓ --- ## ⚠️ Common Mistakes ### Mistake 1: Testing AT the Critical Point ❌ Don't test $f'(c)$ where $c$ is the critical point ✅ Test points on either SIDE of $c$ ### Mistake 2: Wrong Sign Interpretation The sign of $f'$, not $f$, determines increasing/decreasing! ### Mistake 3: Forgetting to Check All Critical Points Each critical point needs its own analysis. ### Mistake 4: Not Using Factored Form Factor $f'(x)$ completely to make sign analysis easier! --- ## First Derivative Test vs. Other Methods ### When to Use the First Derivative Test ✅ **Always works** (if derivative exists on either side) ✅ Good for understanding overall behavior ✅ Tells you increasing/decreasing intervals ✅ Works when second derivative is hard to compute ### When to Use the Second Derivative Test (coming next!) ✅ Faster for simple cases ✅ Only need to evaluate at one point ❌ Fails if $f''(c) = 0$ ❌ Doesn't give increasing/decreasing info --- ## Quick Reference Chart | Sign Change | Result | |-------------|--------| | + → − | Local MAX ⬆️⬇️ | | − → + | Local MIN ⬇️⬆️ | | + → + | Neither (increasing through) | | − → − | Neither (decreasing through) | --- ## 📝 Practice Strategy 1. **Find and factor** $f'(x)$ completely 2. **Mark critical points** on a number line 3. **Test one point** in each interval (use easy numbers) 4. **Record signs** carefully: + for positive, − for negative 5. **Look for sign changes**: + to − is max, − to + is min 6. **State intervals**: Where increasing? Where decreasing? 7. **Calculate $f$ values** at extrema for complete answer