Lecture-11

PART - 1

Applications of Derivatives Lecture 11(i)
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Topics discussed in this lecture:

Uses and difference between a function and an equation

Difference between maxima / minima and maximum / minimum.

How to study NCERT exercise 6.5 maxima and minima

Different types of maxima / minima or maximum / minimum (Global maximum, global minimum, absolute maximum, absolute minimum, local maximum, local minimum)

Questions discussed in this lecture

Question 1. Find the maximum and minimum values, if any, of the following functions given by
(i) f(x) = (2x – 1)^2 + 3
(ii) f(x) = 9x^2 + 12x + 2
(iii) f(x) = -(x – 1)^2 + 10
(iv) g(x) = x^3 + 1

Question 2. Find the maximum and minimum values, if any, of the following functions given by
(i) f(x) = |x + 2| – 1
(ii) g(x) = – |x + 1| + 3
(iii) h(x) = \sin (2x) + 5
(iv) f(x) = |\sin 4x + 3|
(v) h(x) = x + 1, x \in (-1, 1)

PART - 2

Applications of Derivatives Lecture 11(ii)
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Topics discussed in this lecture:

Meaning of Absolute Maximum and Absolute Minimum. Method to find Absolute Maxima and Absolute Minima

 

Questions discussed in this lecture

Question 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 \in [-2, 2]
(ii) f(x) = \sin x + \cos x, x \in [0, \pi]
(iii) f(x) = 4x – \frac{1}{2}x^2, x \in \left[-2, \frac{9}{2} \right]
(iv) f(x) = (x – 1)^2 + 3, x \in [-3, 1]

Question 7. Find both the maximum value and the minimum value of 3x4
– 8x3 + 12x2 – 48x + 25 on the interval [0, 3].

Question 8. At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?

Question 9. What is the maximum value of the function sin x + cos x?

Question 10. Find the maximum value of 2x3 – 24x + 107 in the interval [1, 3]. Find the
maximum value of the same function in [–3, –1].

Question 11. It is given that at x = 1, the function x4– 62x2 + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a.

Question 12. Find the maximum and minimum values of x + sin 2x on [0, 2π].

PART - 3

Lecture-11 1
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Topics discussed in this lecture:

Meaning of Local maxima, local minima, local maximum and local minimum
First derivative test for local maxima and local minima

Questions discussed in this lecture

Question 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
(ii) g(x) = x^3 – 3x
(iii) h(x) = \sin x + \cos x , 0<x<\frac{\pi}{2}
(iv) f(x) = \sin x – \cos x , 0<x<2\pi
(v) f(x) = x^3 – 6x^2 + 9x + 15
(vi) g(x) = \frac{x}{2} + \frac{2}{x},  x>0
(vii) g(x) = \frac{1}{x^2 + 2}
(viii) f(x) = x \sqrt{1 – x}, 0<x<1

Question 4. Prove that the following functions do not have maxima or minima:
(i) f(x) = e^x
(ii) g(x) = \log x
(iii) h(x) = x^3 + x^2 + x + 1

PART - 4

Applications of Derivatives Lecture 11(iv)
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Topics discussed in this lecture:

Meaning of Local maxima, local minima, local maximum and local minimum
Second derivative test for local maxima and local minima

Questions discussed in this lecture

Question 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
(ii) g(x) = x^3 – 3x
(iii) h(x) = \sin x + \cos x , 0<x<\frac{\pi}{2}
(iv) f(x) = \sin x – \cos x , 0<x<2\pi
(v) f(x) = x^3 – 6x^2 + 9x + 15
(vi) g(x) = \frac{x}{2} + \frac{2}{x},  x>0
(vii) g(x) = \frac{1}{x^2 + 2}
(viii) f(x) = x \sqrt{1 – x}, 0<x<1

PART - 5

Lecture-11 2
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Topics discussed in this lecture:

In this lecture, I am explaining and giving proof of “local maximum value can be greater than local minimum value”.

Question discussed in this lecture

Question. Show that the local maximum value of x + \frac{1}{x} is less than local minimum value.

 

PART - 6

Applications of Derivatives Lecture 11(vi)
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Topics discussed in this lecture:

In this lecture I am discussing why we use concepts of Local Maxima and Local Minima to solve word problems in chapter 6 Applications of derivatives through question 17 of exercise 6.5

Question discussed in this lecture

Question 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.

PART - 7

Lecture-11 3
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Topics discussed in this lecture:

In this lecture I am discussing word problems in chapter 6 Applications of derivatives based on Local Maxima and Local Minima.

Question discussed in this lecture

Question 13. Find two numbers whose sum is 24 and whose product is as large as possible.

Question 14. Find two positive numbers x and y such that x + y = 60 and xy3
is maximum.

Question 15. Find two positive numbers x and y such that their sum is 35 and the product x2y5 is a maximum.

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

PART - 8

Applications of Derivatives Lecture 11(viii)
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Topics discussed in this lecture:

In this lecture I am discussing word problems in chapter 6 Applications of derivatives based on Local Maxima and Local Minima.

Question discussed in this lecture

Question 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 ?

Question 19. Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

PART - 9

Applications of Derivatives Lecture 11(ix)
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Topics discussed in this lecture:

In this lecture I am discussing word problems in chapter 6 Applications of derivatives based on Local Maxima and Local Minima.

Question discussed in this lecture

Question 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.

Question 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?

Question 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?

PART - 10

Applications of Derivatives Lecture 11(x)
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Topics discussed in this lecture:

In this lecture I am discussing word problems in chapter 6 Applications of derivatives based on Local Maxima and Local Minima.

Question discussed in this lecture

Question 23. Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is \frac{8}{27} of the volume of the sphere.

Question 24. Show that the right circular cone of least curved surface and given volume has an altitude equal to \sqrt{2} time the radius of the base.

PART - 11

Applications of Derivatives Lecture 11(xi)
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Topics discussed in this lecture:

In this lecture I am discussing word problems in chapter 6 Applications of derivatives based on Local Maxima and Local Minima.

Question discussed in this lecture

Question 25. Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is \tan^{-1} \sqrt{2}.

Question 26. Show that semi-vertical angle of right circular cone of given surface area and maximum volume is \sin^{-1} \left( \frac{1}{3} \right) .

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