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Maxima and Minima
Maxima and Minima - A function f(x) is said to have a local maximum at x = a, if $ is a neighbourhood I of 'a', such that f(a) f(x) for all x I The number f(a) is called the local maximum of f(x). The point a is called the point of maximum. Note that when 'a' is the point of loc..
Maxima and Minima - A function f(x) is said to have a local maximum at x = a, if $ is a neighbourhood I of 'a', such that f(a) f(x) for all x I The number f(a) is called the local maximum of f(x). The point a is called the point of maximum. Note that when 'a' is the point of loc..Maxima and Minima
A function f(x) is said to have a local maximum at x = a, if $ is a neighbourhood I of 'a', such that f(a) f(x) for all x I The number f(a) is called the local maximum of f(x). The point a is called the point of maximu..
A function f(x) is said to have a local maximum at x = a, if $ is a neighbourhood I of 'a', such that f(a) f(x) for all x I The number f(a) is called the local maximum of f(x). The point a is called the point of maximu..Conclusion Differentiation
Conclusion Differentiation - In this chapter, we have studied various techniques of differentiation. Also, we have studied the method of obtaining higher order derivatives of functions which is useful in maxima and minima problems.In this chapter, we have studied various techniq..
Conclusion
We have studied various techniques of differentiation. Also, we have studied the method of obtaining higher order derivatives of functions which is useful in maxima and minima problems...
Example:
Find the local maxima and local minima of the function f (x) = 2x 3 - 21x 2 + 36x - 20. Find also the local maximum and local minimum value..
Conclusion
In this chapter we have learnt the application of derivatives to rate measure, also we have used the geometrical measurement of to find the equations of the tangent and normal to a curve at any point on the curve, angle of intersection of the curves. The derivatives also help in examining the behav..
In this chapter we have learnt the application of derivatives to rate measure, also we have used the geometrical measurement of to find the equations of the tangent and normal to a curve at any point on the curve, angle of intersection of the curves. The derivatives also help in examining the behav..Summary
1. If m = 0 the tangent at (x 1 , y 1 ) is parallel to x-axis. 2. If m 1 = m 2 the curves touch each other. 3. Rolle's theorem 4. Langrange's Mean Value theorem 5. Maxima and Minima..
Solution:
f (x) = x 3 - 6x 2 + 9x + 15 f ' (x) = 3x 2 -12x + 9 = 3(x 2 - 4x + 3) = 3 (x - 1) (x - 3) Thus x = 1 and x = 3 are the only points which could be the points of local maxima or local minima. Let us examine for x=1 When x<1 (slightly less than 1) f '(x) = 3 (x - 1) (x ..
f (x) = x 3 - 6x 2 + 9x + 15 f ' (x) = 3x 2 -12x + 9 = 3(x 2 - 4x + 3) = 3 (x - 1) (x - 3) Thus x = 1 and x = 3 are the only points which could be the points of local maxima or local minima. Let us examine for x=1 When x<1 (slightly less than 1) f '(x) = 3 (x - 1) (x ..Theorem 2:
(First Derivative Test) Let f (x) be a real valued differentiable function. Let a be a point on an interval I such that f '(a) = 0. (a) a is a local maxima of the function f (x) if i) f (a) = 0 ii) f (x) changes sign from positive to negative as x increases through a. That is, f (x) > ..
(First Derivative Test) Let f (x) be a real valued differentiable function. Let a be a point on an interval I such that f '(a) = 0. (a) a is a local maxima of the function f (x) if i) f (a) = 0 ii) f (x) changes sign from positive to negative as x increases through a. That is, f (x) > ..Ratio of Intensities at Maxima and Minima
It has already been proved that the intensity of a wave is proportional to the square of the amplitude. At a point where constructive interference has occurred, the intensity will be maximum and the amplitudes of the two waves will have added. If A 1 and A 2 are the amplitudes of the individual wav..
It has already been proved that the intensity of a wave is proportional to the square of the amplitude. At a point where constructive interference has occurred, the intensity will be maximum and the amplitudes of the two waves will have added. If A 1 and A 2 are the amplitudes of the individual wav..See what our Users say :
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