Maxima & Minima — First and second derivative test
Applications of Derivatives
15
JEE Qs
8%
Hard
75
min
Prioritize understanding the First Derivative Test as it's more robust and serves as a fallback when the Second Derivative Test is inconclusive or computationally intensive.
🧮 Key Formulas
✅ Key Points for JEE
- 1Critical points (where f'(x)=0 or f'(x) is undefined) are the only candidates for local extrema.
- 2The First Derivative Test is universally applicable and works even when the Second Derivative Test fails (f''(x)=0).
- 3The Second Derivative Test is often quicker but is inconclusive if f''(x)=0 at a critical point.
- 4To find global maxima/minima on a closed interval [a,b], evaluate f(x) at all critical points in (a,b) and at the endpoints a and b. The largest value is global maximum, smallest is global minimum.
- 5A point where f'(x)=0 but f'(x) does not change sign (e.g., f(x)=x^3 at x=0) is a point of inflection, not an extremum.
⚠️ Common Mistakes
- ✕Forgetting to check the function values at the endpoints of the interval when finding global maxima/minima.
- ✕Incorrectly interpreting the sign change of f'(x) or making sign errors during calculations.
- ✕Assuming that f''(c)=0 means no extremum exists at c; it only means the Second Derivative Test is inconclusive, and the First Derivative Test must be applied.
- ✕Not considering points where f'(x) is undefined as potential critical points.
📝 Practice Questions
See allQ10.Let the function f(x) = (x2 + 1) x2 −ax + 2 + cos |x| be not differentiable at the two points x = α = 2 and x = β . Then the distance of the point (α, β) from the line 12x + 5y + 10 = 0 is equal to : (1) 5 (2) 4 (3) 3 (4) 2
Q24.Let the function, f(x) = {−3ax2a2 + bx,−2, xx <⩾11 be differentiable for all x ∈R, where a > 1, b ∈R. If the area of the region enclosed by y = f(x) and the line y = −20 is α + β√3, α, β ∈Z , then the value of α + β is ________
Q8. Let f(x) = ∫x20 t2−8t+15et dt, respectively, are : (1) 2 and 3 (2) 2 and 2 (3) 3 and 2 (4) 1 and 3
Q23.If the set of all values of a, for which the equation 5x3 −15x −a = 0 has three distinct real roots, is the interval (α, β), then β −2α is equal to ______
Q13.A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm , the ice-cream melts at the rate of 81 cm3/min and the thickness of the ice-cream layer decreases at the rate of 1 cm/min. The surface area (in cm2 ) of the chocolate ball (without the ice- 4π cream layer) is : (1) 196π (2) 256π (3) 225π (4) 128π
Q20.Let →a = ^i + 2^j + 3^k,→b = 3^i + ^j −^k and →c be three vectors such that →c is coplanar with →a and →b. If the vector →C is perpendicular to →b and →a ⋅→c = 5, then |→c| is equal to (1) √116 (2) 3√21 (3) 16 (4) 18
NCERT Chapters
- Class 12 Mathematics Ch 6: Applications of Derivatives