Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
Chapter 7 — Exercise Solutions Guide
This chapter uses derivatives to study curve behaviour. Always find the derivative first and then decide the type of problem.
How to Use This Page
1. Start with Key Concepts to revise the main theorems and definitions. 2. Attempt MCQs, Fill in the Blanks, and True/False to test your recall. 3. Try Match the Following to link each concept with its formula. 4. Use Answer Briefly (2-mark) for short-answer practice. 5. Study Answer in Detail (5-mark) for step-by-step solutions that board examiners expect. 6. Review Common Mistakes before every mock test. 7. Check the Exam Checklist the night before your exam.
Key Concepts (Quick Recap)
| Term | Definition |
|---|---|
| Derivative (dy/dx) | Rate of change of y with respect to x; gives the slope of the tangent |
| Tangent | Line touching the curve at one point; slope = dy/dx at that point |
| Normal | Line perpendicular to the tangent at the point of contact; slope = −1/(dy/dx) |
| Increasing function | f'(x) > 0 on an interval means the function is rising |
| Decreasing function | f'(x) < 0 on an interval means the function is falling |
| Stationary point | A point where f'(x) = 0; could be a maximum, minimum, or point of inflection |
| Local maximum | f'(c) = 0 and f''(c) < 0; function value is highest in a small neighbourhood |
| Local minimum | f'(c) = 0 and f''(c) > 0; function value is lowest in a small neighbourhood |
| Rolle's theorem | If f is continuous on [a,b] and differentiable on (a,b) with f(a) = f(b), then f'(c) = 0 for some c in (a,b) |
| Lagrange's MVT | If f is continuous on [a,b] and differentiable on (a,b), then f'(c) = [f(b)−f(a)]/(b−a) for some c in (a,b) |
| Point of inflection | Where f''(x) = 0 and the concavity changes |
| Concave up | f''(x) > 0; curve opens upward like a cup |
| Concave down | f''(x) < 0; curve opens downward like a cap |
| Angle between two curves | Angle between their tangents at the point of intersection |
Formula Recap
| Question type | Key idea |
|---|---|
| Tangent slope | dy/dx at the point |
| Normal slope | -1 / tangent slope |
| Increasing function | f'(x) > 0 |
| Decreasing function | f'(x) < 0 |
| Stationary point | f'(x) = 0 |
| Maxima/minima test | Use f''(x) |
I. Choose the Best Answer (MCQs)
1. The slope of the tangent to y = x² at x = 3 is:
(dy/dx = 2x; at x = 3, slope = 6.)
- (a) 3
- (b) 4
- (c) 6
- (d) 9
2. If f'(x) > 0 for all x in (a, b), then f(x) is:
(Positive derivative means the function is increasing.)
- (a) Constant
- (b) Decreasing
- (c) Increasing
- (d) Neither
3. At a local maximum, f''(x) is:
(f''(x) < 0 confirms a local maximum.)
- (a) Positive
- (b) Negative
- (c) Zero
- (d) Undefined
4. Rolle's theorem requires which condition?
(Rolle's theorem requires f(a) = f(b) among other conditions.)
- (a) f(a) = 0 and f(b) = 0
- (b) f(a) = f(b)
- (c) f'(a) = f'(b)
- (d) f(a) > f(b)
5. The equation of the normal to y = x² at (1, 1) is:
(dy/dx = 2x; slope at (1,1) = 2; normal slope = −1/2; equation: y − 1 = −1/2(x − 1) → 2y − 2 = −x + 1 → x + 2y = 3.)
- (a) y = 2x
- (b) y = 2x − 1
- (c) 2y = −x + 3
- (d) x + 2y = 3
6. If f(x) = x³ − 3x, the critical points are:
(f'(x) = 3x² − 3 = 0 → x² = 1 → x = ±1.)
- (a) x = 0
- (b) x = ±1
- (c) x = ±3
- (d) x = ±√3
7. Lagrange's Mean Value Theorem states that there exists c in (a, b) such that:
(Lagrange's MVT gives the slope of the secant line equals the derivative at some point.)
- (a) f'(c) = [f(b) − f(a)]/(b − a)
- (b) f'(c) = 0
- (c) f(c) = [f(b) − f(a)]/(b − a)
- (d) f''(c) = 0
8. A function f is decreasing on (a, b) if:
(Negative derivative means decreasing.)
- (a) f'(x) > 0 for all x in (a, b)
- (b) f'(x) < 0 for all x in (a, b)
- (c) f'(x) = 0 for all x in (a, b)
- (d) f''(x) < 0 for all x in (a, b)
9. The point of inflection for f(x) = x³ is at:
(f''(x) = 6x = 0 at x = 0 and concavity changes.)
- (a) x = 1
- (b) x = −1
- (c) x = 0
- (d) x = 3
10. If the angle between two curves at a point of intersection is 90°, the curves are:
(Curves intersecting at right angle are called orthogonal.)
- (a) Parallel
- (b) Orthogonal
- (c) Tangent
- (d) Identical
II. Fill in the Blanks
f(x) = 5x³ + 2x is ______.y = sin x at x = π/2 is ______.f'(x) = 0 at x = c, then x = c is called a ______.f''(c) must be ______.f(x) = cos x, then the function is decreasing on the interval ______.f'(c) = 0 and f''(c) < 0, then f has a ______ at x = c.III. True or False
1. If f'(x) = 0 at some point, the function must have a maximum or minimum there. False. Correction: f'(x) = 0 gives a stationary point, but it could also be a point of inflection. Use f''(x) to classify.
2. A normal to a curve is always perpendicular to the tangent at the same point. True.
| 3. Rolle's theorem is applicable to f(x) = | x | on [−1, 1]. False. Correction: | x | is not differentiable at x = 0, so Rolle's theorem does not apply on [−1, 1]. |
|---|
4. If f''(x) > 0 on an interval, the function is concave up on that interval. True.
5. The slope of the tangent to y = x³ at x = 0 is 3. False. Correction: dy/dx = 3x²; at x = 0, slope = 0.
6. Lagrange's MVT guarantees at least one point where the tangent is parallel to the secant line. True.
7. A function can be both increasing and decreasing at the same point. False. Correction: At any specific point, a function is either increasing, decreasing, or stationary in its neighbourhood, not both.
8. The absolute maximum of a continuous function on a closed interval always exists. True. (By the Extreme Value Theorem.)
IV. Match the Following
| Column A | Column B | ||
|---|---|---|---|
| 1. f'(x) > 0 on (a, b) | (d) Function is increasing | ||
| 2. f'(x) < 0 on (a, b) | (e) Function is decreasing | ||
| 3. f'(c) = 0 and f''(c) < 0 | (a) Local maximum at x = c | ||
| 4. f'(c) = 0 and f''(c) > 0 | (f) Local minimum at x = c | ||
| 5. Rolle's theorem condition | (g) f(a) = f(b) | ||
| 6. Lagrange's MVT | (b) f'(c) = [f(b) − f(a)]/(b − a) | ||
| 7. Angle between curves | (c) tan⁻¹ | (m1 − m2)/(1 + m1m2) | |
| 8. Slope of normal | (h) −1/f'(x) |
V. Answer Briefly (2-mark)
1. Find the intervals on which f(x) = x² − 4x + 6 is increasing and decreasing.
f is decreasing on (−∞, 2) and increasing on (2, ∞).
2. Verify Rolle's theorem for f(x) = x² − 4 on [−2, 2].
Rolle's theorem is verified with c = 0.
3. Find the equation of the tangent to y = x³ at (1, 1).
The tangent equation is y = 3x − 2.
4. Find the equation of the normal to y = x² at (2, 4).
The normal equation is x + 4y = 18.
5. Determine whether f(x) = x³ − 3x + 5 is increasing or decreasing at x = 2.
Since f'(2) > 0, the function is increasing at x = 2.
VI. Answer in Detail (5-mark)
1. Find the local maximum and local minimum values of f(x) = x³ − 6x² + 9x + 15.
Step 1: Find the first derivative.
Step 2: Find critical points.
Step 3: Find the second derivative.
Step 4: Classify each critical point.
Local maximum = 19 at x = 1; Local minimum = 15 at x = 3.
2. Verify Lagrange's Mean Value Theorem for f(x) = x² on [1, 3] and find the value of c.
Step 1: Check conditions.
Step 2: Apply MVT.
Step 3: Verify.
Lagrange's MVT is verified with c = 2.
3. Find the equation of the tangent and normal to the curve y = x³ − 2x + 3 at the point (1, 2).
Step 1: Find the derivative (tangent slope).
Step 2: Equation of tangent.
Step 3: Equation of normal (slope = −1/m = −1).
The tangent is y = x + 1 and the normal is x + y = 3.
4. Determine the intervals on which f(x) = 2x³ − 3x² − 36x + 7 is increasing and decreasing.
Critical points: x = 3 and x = −2.
Intervals: (−∞, −2), (−2, 3), (3, ∞)
f is increasing on (−∞, −2) ∪ (3, ∞) and decreasing on (−2, 3).
5. A rectangular sheet of metal 40 cm by 20 cm is to be made into an open box by cutting equal squares from each corner and bending up the sides. Find the size of the square to be cut out so that the volume is maximum.
Let x be the side of the square cut from each corner.
Step 1: Differentiate and find critical point.
Since x must be less than 10 (width/2), we take the smaller root:
Step 2: Verify maximum with second derivative.
Since the second derivative is negative, V is maximum at x ≈ 4.23 cm.
The square to be cut out has side approximately 4.23 cm for maximum volume.
Tangent and Normal Pattern
1. Differentiate the given curve. 2. Substitute the given point to get tangent slope. 3. Use point-slope form:
4. For normal, use slope -1/m.
Maxima and Minima Pattern
1. Find f'(x). 2. Solve f'(x) = 0. 3. Find f''(x). 4. If f''(a) < 0, maximum at x = a. 5. If f''(a) > 0, minimum at x = a.
Worked Pattern
If f(x) = x² - 4x + 5:
So minimum occurs at x = 2.
Worked Examples
Example 1 — Slope of a tangent
Find the slope of the tangent to y = x³ at x = 2.
Example 2 — Maxima and minima
Find the local maximum and minimum of f(x) = x³ − 3x + 2.
Local max value 4 at x = −1; local min value 0 at x = 1.
Example 3 — Equation of a tangent line
Find the tangent to y = x² at the point (1, 1).
Common Mistakes to Avoid
- ❌ Forgetting to substitute point after differentiating → ✅ Always evaluate dy/dx at the given point to get the slope
- ❌ Writing normal slope same as tangent slope → ✅ Normal slope = −1/(tangent slope); they are reciprocals with a sign change
- ❌ Stopping at critical point without classifying maximum or minimum → ✅ Always use f''(x) test or sign chart to classify
- ❌ Not checking interval for increasing/decreasing questions → ✅ Use sign chart of f'(x) across critical points to determine intervals
- ❌ Confusing Rolle's theorem with Lagrange's MVT → ✅ Rolle's requires f(a) = f(b); Lagrange's does not require this condition
- ❌ Writing f'(c) = 0 as the conclusion of Lagrange's MVT → ✅ Lagrange's conclusion is f'(c) = [f(b) − f(a)]/(b − a), not f'(c) = 0
- ❌ Assuming f'(x) = 0 always means maximum or minimum → ✅ f'(x) = 0 could also mean a point of inflection; verify with f''(x)
- ❌ Forgetting to verify conditions of Rolle's/MVT before applying → ✅ Check continuity and differentiability conditions first
Exam Checklist
- Derivative is correct (use product rule, quotient rule, or chain rule as needed)
- Slope is evaluated at the specific point given in the question
- Tangent equation uses point-slope form y − y₁ = m(x − x₁)
- Normal slope = −1/m where m is tangent slope
- Critical points found by solving f'(x) = 0
- Maximum/minimum classified using f''(x) test
- Intervals for increasing/decreasing determined by sign chart of f'(x)
- Rolle's theorem: verify f(a) = f(b), continuity, and differentiability
- Lagrange's MVT: verify continuity and differentiability, then find c
- Final equations simplified and written in the required form
- Max/min application problems: set up V or A as function of one variable first