NCERT Solutions for Class 12 Mathematics Chapter 6: Applications of Derivatives – Free PDF Download

Complete, verified solutions to the NCERT Class 12 Maths Chapter 6 (Applications of Derivatives) exercise questions. These Class 12 Mathematics Chapter 6 solutions are also useful as quick revision notes before exams.

NCERT Exercise Solutions

6.1 Find the rate of change of the area of a circle with respect to its radius r when r=3cm and r=4cm.
Ans: A=πr², dA/dr=2πr. At r=3: dA/dr=6π cm²/cm. At r=4: dA/dr=8π cm²/cm.

6.2 The volume of a cube is increasing at a rate of 8 cm³/s. How fast is the surface area increasing when the length of an edge is 12cm?
Ans: V=x³, dV/dt=3x²(dx/dt)=8, so at x=12: dx/dt=8/(3×144)=1/54 cm/s. S=6x², dS/dt=12x(dx/dt)=12(12)(1/54)=144/54=8/3 cm²/s.

6.3 A stone is dropped into a quiet lake and waves move in circles at a speed of 5 cm/s. At the instant when the radius of the circular wave is 8cm, how fast is the enclosed area increasing?
Ans: dr/dt=5 cm/s. A=πr², dA/dt=2πr(dr/dt)=2π(8)(5)=80π cm²/s.

6.4 Find the intervals in which the function f(x)=2x³-3x²-36x+7 is (a) increasing (b) decreasing.
Ans: f′(x)=6x²-6x-36=6(x²-x-6)=6(x-3)(x+2). f′(x)=0 at x=-2,3. Testing sign in intervals: increasing on (-∞,-2)∪(3,∞), decreasing on (-2,3).

6.5 Find the maximum and minimum values, if any, of f(x)=(2x-1)²+3.
Ans: f′(x)=4(2x-1)=0 ⇒ x=1/2. f″(x)=8>0, so x=1/2 is a minimum point. Minimum value = f(1/2)=0+3=3. No maximum (function →∞ as x→±∞).

Frequently Asked Questions

What is the second derivative test for local extrema?
If f′(c)=0 and f″(c)>0, c is a local minimum. If f″(c)<0, c is a local maximum. If f″(c)=0, the test is inconclusive.

What is the condition for a function to be increasing on an interval?
f′(x) ≥ 0 for all x in that interval (strictly increasing if f′(x)>0).

Written by Satish

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