The Relationship Between f(x), f'(x), and f''(x) for Differentiability
Understand how functions and their derivatives connect geometrically and analytically for JEE success.
Why This Relationship Matters for JEE
Understanding how f(x), f'(x), and f''(x) relate is crucial for solving 3-4 questions in every JEE paper. These relationships help you:
- Analyze function behavior without graphing
- Solve maxima/minima problems efficiently
- Understand curve sketching concepts deeply
- Tackle differentiability questions confidently
🎯 Quick Navigation
1. f'(x) - The First Derivative: What It Tells Us
When f'(x) > 0
📈 Geometric Interpretation
The function is increasing at that point. The tangent line has positive slope.
🎯 JEE Example
For $f(x) = x^2$, $f'(x) = 2x$. When $x > 0$, $f'(x) > 0$, so the function is increasing.
When f'(x) < 0
📉 Geometric Interpretation
The function is decreasing at that point. The tangent line has negative slope.
🎯 JEE Example
For $f(x) = x^2$, $f'(x) = 2x$. When $x < 0$, $f'(x) < 0$, so the function is decreasing.
When f'(x) = 0
⚖️ Geometric Interpretation
The function has a critical point (could be local max, local min, or point of inflection).
🎯 JEE Example
For $f(x) = x^2$, $f'(x) = 2x = 0$ when $x = 0$. This is a local minimum.
2. f''(x) - The Second Derivative: Curvature Analysis
When f''(x) > 0
⎰ Geometric Interpretation
The function is concave up (convex). The curve lies above its tangent lines.
🎯 JEE Example
For $f(x) = x^2$, $f''(x) = 2 > 0$ for all x. The parabola is always concave up.
When f''(x) < 0
⎱ Geometric Interpretation
The function is concave down (concave). The curve lies below its tangent lines.
🎯 JEE Example
For $f(x) = -x^2$, $f''(x) = -2 < 0$ for all x. The parabola is always concave down.
When f''(x) = 0
🔄 Geometric Interpretation
Possible point of inflection - where concavity changes.
🎯 JEE Example
For $f(x) = x^3$, $f''(x) = 6x = 0$ when $x = 0$. This is a point of inflection.
3. Geometric Relationships: Putting It All Together
The Complete Picture
| f'(x) | f''(x) | Function Behavior | JEE Significance |
|---|---|---|---|
| Positive | Positive | Increasing, Concave Up | Rapid growth, exponential-like behavior |
| Positive | Negative | Increasing, Concave Down | Slowing growth, approaching horizontal asymptote |
| Negative | Positive | Decreasing, Concave Up | Slowing decrease, approaching minimum |
| Negative | Negative | Decreasing, Concave Down | Rapid decrease, exponential decay |
💡 Memory Aid
"Positive f'(x) means going up, Positive f''(x) means smiling up"
Think of f''(x) > 0 as a smile (concave up) and f''(x) < 0 as a frown (concave down).
4. Differentiability: Key Conditions and Common Pitfalls
When is a Function Differentiable?
✅ Differentiable at x = a if:
- f(x) is continuous at x = a
- Left-hand derivative = Right-hand derivative
- No sharp corners or cusps
- No vertical tangents
❌ NOT Differentiable at x = a if:
- Discontinuity at x = a
- Sharp corner (like |x| at x = 0)
- Vertical tangent (like $x^{1/3}$ at x = 0)
- Cusp point
⚠️ Common JEE Trap
Continuity ≠ Differentiability! A function can be continuous but not differentiable.
Example: $f(x) = |x|$ is continuous at x = 0 but not differentiable there (sharp corner).
🎯 JEE Differentiability Problem
Question: Check differentiability of $f(x) = \begin{cases} x^2 & \text{if } x \leq 1 \\ 2x-1 & \text{if } x > 1 \end{cases}$ at x = 1
Step 1: Check continuity at x = 1
Left limit: $\lim_{x \to 1^-} x^2 = 1$
Right limit: $\lim_{x \to 1^+} (2x-1) = 1$
f(1) = 1. Continuous ✓
Step 2: Check differentiability
Left derivative: $f'(x) = 2x \Rightarrow f'(1^-) = 2$
Right derivative: $f'(x) = 2 \Rightarrow f'(1^+) = 2$
Equal derivatives ✓
Conclusion: Differentiable at x = 1
5. JEE Applications and Problem Solving Strategies
Maxima/Minima Problems
Second Derivative Test
Curve Sketching Strategy
Find critical points (f'(x) = 0)
Determine increasing/decreasing intervals
Find inflection points (f''(x) = 0)
Determine concavity intervals
🚀 Time-Saving JEE Strategy
For multiple choice questions: Often you can eliminate options by checking just the signs of f'(x) and f''(x) without full calculations.
Example: If asked about local maxima/minima, immediately check if f''(x) is positive or negative at critical points.
📋 Quick Reference Table
| Function Property | f'(x) Condition | f''(x) Condition | Geometric Meaning |
|---|---|---|---|
| Increasing | f'(x) > 0 | - | Slope positive |
| Decreasing | f'(x) < 0 | - | Slope negative |
| Local Maximum | f'(x) = 0 | f''(x) < 0 | Peak point |
| Local Minimum | f'(x) = 0 | f''(x) > 0 | Valley point |
| Concave Up | - | f''(x) > 0 | Smile shape |
| Concave Down | - | f''(x) < 0 | Frown shape |
📝 Test Your Understanding
Try these JEE-style problems to apply what you've learned:
1. For $f(x) = x^3 - 3x^2 + 2$, find intervals where:
a) f(x) is increasing
b) f(x) is concave up
c) Local maxima and minima
2. Determine if $f(x) = |x-2| + |x+1|$ is differentiable at x = 2
3. A function has f'(1) = 0 and f''(1) = -3. What can you conclude about f(x) at x = 1?
Mastered the Relationships?
You're now ready to tackle any JEE question involving f(x), f'(x), and f''(x) relationships!