JEE Maths›Application of Derivatives›Question Types

Application of Derivatives - How It Appears in JEE

8 recurring patterns. Learn the pattern, recognize it in 5 seconds, apply the right approach.

Rate of Change Problems

Pattern

Find rate of change of one quantity with respect to another

How to recognize

Problems asking how fast something changes: area of circle increasing, volume of sphere changing, distance varying with time

→ Express the quantity as a function of the given variable. Differentiate with respect to the appropriate variable (usually time). Substitute the given values.

Tangent and Normal Equations

Pattern

Find the equation of tangent or normal to a curve at a given point

How to recognize

Problems involving tangent/normal to y = f(x), parametric curves, or implicit curves at a specified point or with a given slope condition

→ Find dy/dx. Evaluate at the given point to get slope. Use point-slope form. For normal, use negative reciprocal of tangent slope.

Intervals of Increase/Decrease

Pattern

Determine where a function is increasing or decreasing

How to recognize

Questions asking for intervals, range of parameter for monotonicity, or proving a function is always increasing/decreasing

→ Find f'(x). Solve f'(x) = 0 for critical points. Use number line or sign analysis to determine where f'(x) > 0 (increasing) or f'(x) < 0 (decreasing).

Local Maxima and Minima

Pattern

Find local extreme values using first or second derivative test

How to recognize

Questions asking for local/relative maxima or minima, critical points, or nature of stationary points

→ Find critical points from f'(x) = 0 and where f'(x) DNE. Apply first derivative test (sign change) or second derivative test (sign of f''). Evaluate f at critical points.

Absolute Max/Min on Closed Interval

Pattern

Find global maximum or minimum of f on [a, b]

How to recognize

Questions specifying a closed interval and asking for greatest/least value, absolute/global extrema

→ Find all critical points in (a, b). Evaluate f at each critical point and at endpoints a and b. Compare all values to identify the absolute max and min.

Optimization Problems

Pattern

Maximize or minimize a quantity subject to constraints

How to recognize

Word problems: maximize area, minimize cost, largest volume, shortest distance, most profit. A constraint equation relates the variables.

→ Express the objective function in one variable using the constraint. Differentiate, set f'(x) = 0, find critical points. Verify max/min using second derivative or boundary analysis.

Rolle's Theorem and LMVT Applications

Pattern

Apply Rolle's theorem or Lagrange's Mean Value Theorem

How to recognize

Questions asking to verify conditions, find the value of c, or prove existence of a point where f'(c) equals a specific value on an interval

→ Check all conditions: continuity on [a, b], differentiability on (a, b), and f(a) = f(b) for Rolle's. For LMVT: f'(c) = [f(b) - f(a)]/(b - a). Solve for c in (a, b).

Number of Roots Using Rolle's Theorem

Pattern

Determine the number of real roots of an equation using Rolle's theorem

How to recognize

Questions asking to prove exactly n roots exist, or find how many solutions an equation has in a given interval

→ If f(x) has n roots, then f'(x) has at least (n-1) roots by Rolle's theorem. Work backwards: analyze f'(x) to bound the number of roots of f(x). Use IVT for existence and Rolle's for upper bound.

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