Finding maximum and minimum values of functions using derivatives and the first/second derivative test.
## Core concept
Maxima and minima are the highest and lowest points of a function within a given interval or domain. They are critical for optimization problems in business (cost minimization, profit maximization, production scheduling).
Key principle: At turning points (local maxima/minima), the first derivative equals zero.
- Local maximum: Function value is greater than all nearby values
- Local minimum: Function value is less than all nearby values
- Global maximum/minimum: Highest/lowest value over the entire domain
## Formula / rule
Step 1: Find critical points - Differentiate the function: f'(x) = dy/dx - Set f'(x) = 0 and solve for x - These x-values are candidates for maxima/minima
Step 2: Classify using first derivative test - Check sign of f'(x) just before and after each critical point - If f'(x) changes from positive to negative → local maximum - If f'(x) changes from negative to positive → local minimum - If sign doesn't change → point of inflection (neither max nor min)
Step 3: Classify using second derivative test (faster method) - Find the second derivative: f''(x) = d²y/dx² - Evaluate f''(x) at each critical point - If f''(x) < 0 → local maximum - If f''(x) > 0 → local minimum - If f''(x) = 0 → test is inconclusive; use first derivative test
Step 4: Find actual max/min values - Substitute critical point x-values back into original function f(x) - Also check boundary points if domain is restricted [a, b] - Compare all values to identify global max/min
## Common exam applications
Profit maximization: Given profit P(x) = −2x² + 100x − 500, find production quantity x for maximum profit. - P'(x) = −4x + 100 - Set P'(x) = 0 → x = 25 units - P''(x) = −4 < 0 → maximum - Max profit = P(25) = −2(625) + 2500 − 500 = ₹875
Cost minimization: Minimize average cost AC(x) = x + 100/x for x > 0. - d(AC)/dx = 1 − 100/x² - Set = 0 → x² = 100 → x = 10 - d²(AC)/dx² = 200/x³ = 200/1000 = 0.2 > 0 → minimum
Word problems: Revenue, inventory management, distance-time optimization.
## Common mistakes
- Forgetting to verify critical points – Just solving f'(x) = 0 isn't enough; must confirm max/min using derivatives or boundary check
- Confusing critical points with actual extrema – f'(x) = 0 is necessary but not sufficient; inflection points also satisfy it
- Ignoring boundary points – In closed intervals [a, b], global extrema may occur at endpoints, not just at f'(x) = 0
- Second derivative test fails silently – When f''(x) = 0, students assume "no max/min"; must revert to first derivative test
- Substitution errors – Finding x-coordinate correctly but computing f(x) incorrectly; verify arithmetic twice
- Sign errors in derivatives – Careless differentiation leads to wrong critical points; use product/quotient rules carefully
## Quick checklist for exam
- [ ] Differentiate correctly
- [ ] Solve f'(x) = 0 for all critical points
- [ ] Apply first or second derivative test to classify
- [ ] Substitute back into f(x) to find actual max/min values
- [ ] Check domain boundaries if restricted
- [ ] State answer clearly with units (if word problem)