Class 12 • mathematics • Chapter 6
APPLICATION OF DERIVATIVES
True & False Quiz
Rise. Fall. Optimise.
✓True
✗False
25
Questions
|
Ch.6
Chapter
|
XII
Class
Why True & False for APPLICATION OF DERIVATIVES?
How this format sharpens your conceptual clarity
🔵 Derivatives model rates of change and optimisation — from cost minimisation to maximising area under constraints.
✅ T/F targets increasing/decreasing tests, and distinguishing local extrema from absolute (global) extrema.
🎯 Trap: f′(c) = 0 does NOT guarantee a maximum or minimum at c — it could be a point of inflection.
📋
Read each statement carefully. Click True or False — instant feedback with explanation appears. Submit anytime; unattempted questions are marked Skipped.
Q 1
If a function is increasing on an interval, then its derivative is positive at every point of that interval.
Q 2
If \(f'(x)>0\) for every \(x\) in an interval, then \(f(x)\) is strictly increasing on that interval.
Q 3
If \(f'(x)<0\) for every \(x\) in an interval, then \(f(x)\) is strictly decreasing on that interval.
Q 4
If \(f'(a)=0\), then \(x=a\) must be a point of local maximum or local minimum.
Q 5
If \(f'(x)\) changes from positive to negative at \(x=a\), then \(f\) has a local maximum at \(x=a\).
Q 6
If \(f'(x)\) changes from negative to positive at \(x=a\), then \(f\) has a local minimum at \(x=a\).
Q 7
A stationary point of a differentiable function is necessarily an extremum.
Q 8
If \(f'(x)\geq 0\) throughout an interval, then \(f\) cannot decrease on that interval.
Q 9
The function \(f(x)=x^2\) is decreasing on \((-\infty,0)\) and increasing on \((0,\infty)\).
Q 10
The function \(f(x)=x^3-3x\) has a local maximum at \(x=-1\) and a local minimum at \(x=1\).
Q 11
If \(f'(a)=0\) and \(f''(a)>0\), then \(f\) has a local maximum at \(x=a\).
Q 12
If \(f'(a)=0\) and \(f''(a)<0\), then \(f\) has a local maximum at \(x=a\).
Q 13
If \(f'(a)=0\) and \(f''(a)=0\), then \(x=a\) is neither a maximum nor a minimum.
Q 14
If a differentiable function has a local extremum at an interior point \(a\), then \(f'(a)=0\).
Q 15
The maximum value of a function on a closed interval can occur only at an interior critical point.
Q 16
To find the absolute maximum and minimum of a continuous function on \([a,b]\), it is sufficient to check only the points where \(f'(x)=0\).
Q 17
The function \(f(x)=x+\dfrac{1}{x}\) is increasing on \((1,\infty)\).
Q 18
If \(f'(x)\) is positive at a single point \(x=a\), then \(f\) is increasing throughout the entire domain.
Q 19
For \(f(x)=x^3-6x^2+9x+1\), the points \(x=1\) and \(x=3\) are critical points.
Q 20
If \(f'(x)\) changes sign from positive to negative at \(x=a\), then \(f(a)\) is necessarily the greatest value of \(f\) over its entire domain.
Q 21
For \(f(x)=x^3-3x^2+1\), \(x=0\) is a local maximum and \(x=2\) is a local minimum.
Q 22
If a function is continuous on \([a,b]\) and differentiable on \((a,b)\), then its absolute maximum and minimum must both occur at interior points.
Q 23
For \(f(x)=x^4-4x^2\), the point \(x=0\) is a local maximum even though \(f'(0)=0\).
Q 24
If \(f'(a)=0\), \(f''(a)=0\), and \(f'''(a)\neq0\), then \(x=a\) cannot be a point of local maximum or local minimum.
Q 25
If \(f'(a)=0\) and the first non-zero derivative of \(f\) at \(a\), after the first derivative, is of even order and is positive, then \(f\) has a local minimum at \(x=a\).
Key Takeaways — APPLICATION OF DERIVATIVES
Core facts for CBSE Boards & JEE
1
f is increasing on an interval if f′(x) ≥ 0 there, and decreasing if f′(x) ≤ 0 there.
2
f′(c) = 0 is necessary but NOT sufficient for a local extremum — the second derivative test confirms it.
3
At a local maximum, f″(c) < 0; at a local minimum, f″(c) > 0 (second derivative test).
4
The equation of the tangent to y=f(x) at (x₁,y₁) is y−y₁ = f′(x₁)(x−x₁); the normal is perpendicular to it.
5
Rate of change problems use dV/dt, dA/dt etc. via the chain rule — related rates, not a single fixed derivative.
6
A local maximum value can be smaller than a local minimum value elsewhere on the same curve — "local" is not "global".