Maxima & Minima — First + Second Derivative Test
Applications of Derivatives
75
JEE Qs
28%
Hard
100
min
Always correctly identify all critical points (f'(x)=0 or f'(x) undefined) and systematically apply the appropriate derivative test, remembering to check endpoints for absolute extrema on closed intervals.
🧮 Key Formulas
✅ Key Points for JEE
- 1Local maxima/minima always occur at critical points (where the derivative is zero or undefined) or at endpoints of an interval.
- 2The First Derivative Test is universally applicable for finding local extrema, even when the second derivative does not exist or is zero.
- 3The Second Derivative Test is generally quicker when f''(x) is easy to compute and non-zero at the critical points.
- 4When applying the Second Derivative Test, if f''(c)=0, it is crucial to revert to the First Derivative Test to determine the nature of the critical point.
- 5For absolute (global) maxima and minima on a closed interval [a,b], evaluate the function at all critical points within (a,b) and at the endpoints a and b; the largest/smallest of these values is the absolute maximum/minimum.
⚠️ Common Mistakes
- ✕Confusing local extrema with absolute extrema; absolute extrema require checking endpoints of the interval.
- ✕Incorrectly concluding a local max/min when f''(c)=0 instead of using the First Derivative Test.
- ✕Failing to identify all critical points, especially those where f'(x) is undefined (e.g., for |x|, x^(1/3)).
- ✕Errors in analyzing the sign change of f'(x) for the First Derivative Test, leading to incorrect classification of extrema.
📝 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