Curve Sketching with Derivatives
Derivatives tell us everything about a function's shape:
- First derivative → where it's increasing/decreasing
- Second derivative → where it's concave up/down
First Derivative Test
Second Derivative Test
Analyze f(x) = x³ - 3x
Medium
Find where f(x) = x³ - 3x is increasing, decreasing, and locate max/min.
1
First derivative
f'(x) = 3x² - 3 = 3(x² - 1) = 3(x+1)(x-1)
2
Critical points
f'(x) = 0 when x = -1, 1
3
Sign analysis
x < -1: f'(x) > 0 (increasing)
-1 < x < 1: f'(x) < 0 (decreasing)
x > 1: f'(x) > 0 (increasing)
4
Conclusion
Local max at x = -1: f(-1) = 2
Local min at x = 1: f(1) = -2
Answer:
Local max (−1, 2), local min (1, −2)
Practice
1. If f'(x) > 0 and f''(x) < 0, the function is:
Key Takeaways
f' tells us increasing/decreasing (slope)
f'' tells us concavity (curvature)
Critical points where f' = 0 or undefined
Inflection points where f'' changes sign