Question
Find all local maxima and minima of f(x) = x³ - 3x + 1. Also state the local maximum and minimum values.
This question appeared in the CBSE 2024 Board Exam and is a standard first derivative test problem. Full marks require both the x-coordinates AND the function values.
Solution — Step by Step
Differentiate f(x) = x³ - 3x + 1 term by term:
This tells us the slope of the curve at every point. Where slope = 0, we have a candidate for extremum.
We get two critical points: x = 1 and x = -1. Both must be tested — don’t discard either.
The second derivative test tells us: if f''(x) > 0 at a critical point, it’s a local minimum; if f''(x) < 0, it’s a local maximum.
At x = 1:
At x = -1:
Local minimum value (at x = 1):
Local maximum value (at x = -1):
Final Answer: Local maximum = 3 at x = -1; Local minimum = -1 at x = 1.
Why This Works
The function f(x) = x³ - 3x + 1 is a cubic polynomial, and all cubics have an S-shape. The curve rises steeply, flattens out at a “hump” (local max), dips down to a “valley” (local min), then rises steeply again. The derivative f'(x) captures where this flattening happens — it’s zero exactly at these turning points.
The second derivative test works because f''(x) measures the rate of change of slope. At x = -1, the slope is going from positive (rising) to zero to negative (falling) — that’s a peak. Since the slope is decreasing there, f'' is negative. At x = 1, slope goes from negative to zero to positive — that’s a valley, so f'' is positive.
This logic holds universally: positive f'' means the curve is “concave up” like a bowl (holds water), which is always a minimum. Negative f'' means “concave down” like an arch, which is always a maximum.
Alternative Method
We can use the first derivative sign change test instead — useful when f''(c) = 0 (second derivative test fails).
Check the sign of f'(x) = 3(x² - 1) = 3(x-1)(x+1) in three intervals:
| Interval | Sign of (x-1) | Sign of (x+1) | Sign of f'(x) |
|---|---|---|---|
x < -1 | − | − | + (increasing) |
-1 < x < 1 | − | + | − (decreasing) |
x > 1 | + | + | + (increasing) |
At x = -1: sign changes from + to − → Local Maximum
At x = 1: sign changes from − to + → Local Minimum
Same conclusion, no second derivative needed.
In CBSE board exams, both methods fetch full marks. But the second derivative test is faster when f”(c) ≠ 0 — use it by default and switch to sign change only when f”(c) = 0.
Common Mistake
Stopping after finding critical points. Many students write “x = 1 is minimum and x = -1 is maximum” and stop there — losing 1-2 marks. The question asks for local maximum and minimum values. You must substitute back: f(-1) = 3 and f(1) = -1. Always re-read whether the question wants the x-coordinate (point) or the y-value (value of function).
A second common slip: computing f(-1) as (-1)³ = 1 instead of -1. Odd powers of negative numbers stay negative. (-1)³ = -1, not +1.