# NCERT Solutions for Class 12 Maths Application of Derivatives

Hi Students, Welcome to **Amans Maths Blogs (AMB)**. In this post, you will get the * NCERT Solutions for Class 12 Maths Application of Derivative Exercise 6.5*.

* CBSE Class 12^{th}* is an important school class in your life as you take some serious decision about your career. And out of all subjects, Maths is an important and core subjects. So

*is major role in your exam preparation as it has detailed chapter wise solutions for all exercise. This*

**CBSE NCERT Solutions for Class 12**^{th}Maths**can be downloaded in PDF file. The downloading link is given at last.**

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**NCERT Solutions for Class 12 Maths Application of Derivative Exercise 6.2**## NCERT Solutions for Class 12 Maths Application of Derivatives Exercise 6.5

**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 1.**

Find the maximum and minimum values, if any, of the following functions given by

(i) f (x) = (2x – 1)^{2} + 3

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(ii) f (x) = 9x^{2} + 12x + 2

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(iii) f (x) = – (x – 1)^{2} + 10

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(iv) f (x) = x^{3} + 1

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 2.**

Find the maximum and minimum values, if any, of the following functions given by

(i) f (x) = |x + 2 | – 1

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(ii) g(x) = – | x + 1| + 3

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(iii) h(x) = sin (2x) + 5

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(iv) f (x) = | sin 4x + 3|

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(v) h(x) = x + 1, x ∈ (– 1, 1)

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 3.**

Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:

(i) f (x) = x^{2}

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(ii) g(x) = x^{3} – 3x

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(iii) h(x) = sin x + cos x, 0 < x < π/2.

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(iv) f (x) = sin x – cos x, 0 < x < 2π.

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(v) f (x) = x^{3} – 6x^{2} + 9x + 15

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(vi) g (x) = x/2 + 2/x, x > 0

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(vii) g (x) = 1/(x^{2} + 2)

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(viii) f (x) = x√(1 – x), x > 0

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 4.**

Prove that the following functions do not have maxima or minima

(i) f (x) = e^{x}

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(ii) g(x) = log x

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(iii) h (x) = x^{3} + x^{2} + x +1

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 5.**

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:

(i) f (x) = x^{3}, x ∈ [– 2, 2]

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(ii) f (x) = sin x + cos x , x ∈ [0, π]

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(iii) f (x) = 4x – x^{2}/2 , x ∈ [-2, 9/2]

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(iv) f (x) = (x −1)^{2} + 3 , x ∈ [-3, 1]

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 6.**

Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 – 24x – 18x^{2}

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 7.**

Find both the maximum value and the minimum value of 3x^{4} – 8x^{3} + 12x^{2} – 48x + 25 on the interval [0, 3].

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 8.**

At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 9.**

What is the maximum value of the function sin x + cos x?

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 10.**

Find the maximum value of 2x^{3} – 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [–3, –1].

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 11.**

It is given that at x = 1, the function x^{4} – 62x^{2} + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a.

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 12.**

Find the maximum and minimum values of x + sin 2x on [0, 2π].

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 13.**

Find two numbers whose sum is 24 and whose product is as large as possible.

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 14.**

Find two positive numbers x and y such that x + y = 60 and xy^{3} is maximum.

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 15.**

Find two positive numbers x and y such that their sum is 35 and the product x^{2}y^{5} is a maximum.

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 16.**

Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 17.**

A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible.

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 18.**

A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum ?

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 19.**

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 20.**

Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 21.**

Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 22.**

A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 23.**

Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 8/27 of the volume of the sphere.

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 24.**

Show that the right circular cone of least curved surface and given volume has an altitude equal to √2 time the radius of the base.

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 25.**

Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is tan^{-1}√2.

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 26.**

Show that semi-vertical angle of right circular cone of given surface area and maximum volume is sin^{-1}(1/3).

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Choose the correct answer in the Exercises 27 and 29.

**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 27.**

The point on the curve x^{2} = 2y which is nearest to the point (0, 5) is

(A) (2√2,4) (B) (2√2,0) (C) (0, 0) (D) (2, 2)

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 28.**

For all real values of x, the minimum value of (1 – x + x^{2})/(1 + x + x^{2}) is

(A) 0 (B) 1 (C) 3 (D) 1/3

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**NCERT Solutions for Class 12 Maths Application of Derivatives ****Exercise**** 6.5: Ques No 29.**

The maximum value of [x(x −1) +1]^{1/3}, 0 ≤ x ≤ 1 is

(A) (1/3)^{1/3} (B) 1/2 (C) 1 (D) 0

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