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Chapter 6: Applications of Derivatives

NCERT Solutions for CBSE Class 10 Mathematics — 35 solved questions with detailed explanations.

35

Questions

0

Topics

Solved Questions

Q1. The function f(x) = x³ - 3x + 2 has a local maximum at:

x = 0
x = 1
x = -1
x = 2

Difficulty: Easy-Medium · Topic: Maxima Minima

Solution

f'(x) = 3x²-3 = 0 ⇒ x = ±1. f''(x) = 6x. f''(-1) = -6 < 0, so x=-1 is local maximum.

Q2. The rate of change of area of a circle with respect to radius at r = 5 is:

10π
25π
5π
50π

Difficulty: Easy-Medium · Topic: Rate of Change

Solution

A = πr². dA/dr = 2πr. At r=5: dA/dr = 10π.

Q3. Rolle's theorem requires the function to be:

Continuous on [a,b], differentiable on (a,b), f(a)=f(b)
Only continuous
Only differentiable
Monotonic

Difficulty: Easy-Medium · Topic: Rolle's Theorem

Solution

Rolle's: f continuous on [a,b], differentiable on (a,b), f(a)=f(b) ⇒ ∃c ∈ (a,b) such that f'(c)=0.

Q4. If f'(x) > 0 in interval, then f is:

constant
undefined
increasing
decreasing

Difficulty: Easy-Medium · Topic: Applications of Derivatives

Solution

increasing.

Q5. If f'(x) < 0 in interval, then f is:

constant
zero
decreasing
increasing

Difficulty: Easy-Medium · Topic: Applications of Derivatives

Solution

decreasing.

Q6. At point of inflection:

f''(x) = 0 or changes sign
f'''(x) = 0
f(x) = 0
f'(x) = 0

Difficulty: Easy-Medium · Topic: Applications of Derivatives

Solution

f''(x) = 0 or changes sign.

Q7. LMVT states f'(c) =:

f'(a)+f'(b)
f(b)-f(a)
(f(b)-f(a))/(b-a)
(f(a)+f(b))/2

Difficulty: Easy-Medium · Topic: Applications of Derivatives

Solution

(f(b)-f(a))/(b-a).

Q8. f(x)=1x²+-6x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=2x+-6=0, x=3.0.

Q9. f(x)=1x²+-4x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=2x+-4=0, x=2.0.

Q10. f(x)=1x²+-2x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=2x+-2=0, x=1.0.

Q11. f(x)=1x²+0x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=2x+0=0, x=0.0.

Q12. f(x)=1x²+2x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=2x+2=0, x=-1.0.

Q13. f(x)=1x²+4x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=2x+4=0, x=-2.0.

Q14. f(x)=1x²+6x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=2x+6=0, x=-3.0.

Q15. f(x)=2x²+-6x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=4x+-6=0, x=1.5.

Q16. f(x)=2x²+-4x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=4x+-4=0, x=1.0.

Q17. f(x)=2x²+-2x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=4x+-2=0, x=0.5.

Q18. f(x)=2x²+0x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=4x+0=0, x=0.0.

Q19. f(x)=2x²+2x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=4x+2=0, x=-0.5.

Q20. f(x)=2x²+4x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=4x+4=0, x=-1.0.

Q21. f(x)=2x²+6x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=4x+6=0, x=-1.5.

Q22. f(x)=3x²+-6x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=6x+-6=0, x=1.0.

Q23. f(x)=3x²+-4x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=6x+-4=0, x=0.67.

Q24. f(x)=3x²+-2x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=6x+-2=0, x=0.33.

Q25. f(x)=3x²+0x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=6x+0=0, x=0.0.

Q26. f(x)=3x²+2x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=6x+2=0, x=-0.33.

Q27. f(x)=3x²+4x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=6x+4=0, x=-0.67.

Q28. f(x)=3x²+6x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=6x+6=0, x=-1.0.

Q29. f(x)=4x²+-6x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=8x+-6=0, x=0.75.

Q30. f(x)=4x²+-4x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=8x+-4=0, x=0.5.

Q31. f(x)=4x²+-2x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=8x+-2=0, x=0.25.

Q32. f(x)=4x²+0x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=8x+0=0, x=0.0.

Q33. f(x)=4x²+2x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=8x+2=0, x=-0.25.

Q34. f(x)=4x²+4x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=8x+4=0, x=-0.5.

Q35. f(x)=4x²+6x has minimum at x=

Difficulty: Easy-Medium · Topic: Optimization

Solution

f'(x)=8x+6=0, x=-0.75.

Other Chapters in Mathematics

Ch 1: Relations & Functions Ch 2: Inverse Trig Functions Ch 3: Matrices Ch 4: Determinants Ch 5: Continuity & Differentiability Ch 7: Integrals Ch 8: Applications of Integrals Ch 9: Differential Equations Ch 10: Vector Algebra Ch 11: Three Dimensional Geometry Ch 12: Linear Programming Ch 13: Probability

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