Application of Derivatives

Maths · Class 12

Simulation · Maths · Class 12

Where f rises, falls and turns

"Where f rises, falls and turns" is a free, interactive Class 12 Maths simulation for JEE, built on the lesson "Increasing and decreasing functions" in Application of Derivatives. The short line is the tangent: green while f is rising (f′ > 0), red while it is falling (f′ < 0). The strip under the graph shows the sign of f′ everywhere at once. The point clicks onto each place where f′ = 0. Mark the zeros of f′ and read its sign between them. + then − is a local maximum, − then + is a local minimum, and no change of sign means no extremum. JEE Main asked 47 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).

From the lesson Increasing and decreasing functions in Application of Derivatives. Change the values and watch what happens.

Where f rises, falls and turnsMaths · Class 12

In JEE Main

42 shifts analysed

47 questions from Application of Derivatives across the 42 JEE Main shifts Lumi has analysed (2017–2026).

See the past questions

The idea behind it

NCERT §6.3

  • On an interval I, f is increasing if x₁ < x₂ in I gives f(x₁) ≤ f(x₂), and strictly increasing if it gives f(x₁) < f(x₂). Decreasing and strictly decreasing are defined with the inequalities reversed.
  • A constant function satisfies both ≤ and ≥, so it counts as increasing and as decreasing on any interval, but it is strictly neither.
  • Derivative test: suppose f is differentiable on (a, b) and continuous on the closed interval [a, b]. If f′ > 0 on (a, b), f is increasing on [a, b]; if f′ < 0, f is decreasing; if f′ = 0 throughout, f is constant.
  • The proof uses the mean value theorem: f(x₂) − f(x₁) = f′(c)(x₂ − x₁) for some c between them, so the sign of f′ fixes the sign of the change.
  • Worked example: f(x) = 7x − 3 has f′ = 7 > 0, so it is increasing on R.
  • Worked example: f(x) = x³ − 3x² + 4x has f′ = 3x² − 6x + 4 = 3(x − 1)² + 1, which is always positive, so f is increasing on R.
  • Worked example: cos x has derivative −sin x, which is negative on (0, π) and positive on (π, 2π). So cos x decreases on (0, π) and increases on (π, 2π), and is neither on the whole of (0, 2π).
  • f′ may vanish at single points and f can still be strictly increasing: x³ has f′(0) = 0 yet rises through 0.

Questions

3 answers

What does the "Where f rises, falls and turns" simulation show?
The short line is the tangent: green while f is rising (f′ > 0), red while it is falling (f′ < 0). The strip under the graph shows the sign of f′ everywhere at once. The point clicks onto each place where f′ = 0.
Does JEE Main ask questions from Application of Derivatives?
Yes. JEE Main asked 47 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
Is this simulation free?
Yes. It is free and runs in the browser on a phone or a laptop. It belongs to the lesson "Increasing and decreasing functions" in Application of Derivatives (Class 12 Maths).

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