Advertisements
Advertisements
प्रश्न
f(x) = sin 2x, 0 < x < \[\pi\] .
Advertisements
उत्तर
\[\text { Given }: \hspace{0.167em} f\left( x \right) = \sin 2x\]
\[ \Rightarrow f'\left( x \right) = 2 \cos 2x\]
\[\text { For a local maximum or a local minimum, we must have }\]
\[f'\left( x \right) = 0\]
\[ \Rightarrow 2 \cos 2x = 0\]
\[ \Rightarrow \cos 2x = 0\]
\[ \Rightarrow x = \frac{\pi}{4} or \frac{3\pi}{4}\]
Sincef '(x) changes from positive to negative when x increases through \[\frac{\pi}{4}\], x = \[\frac{\pi}{4}\] is the point of maxima.
The local maximum value of f (x) at x = \[\frac{\pi}{4}\] is given by \[\sin\left( \frac{\pi}{2} \right) = 1\]
Sincef '(x) changes from negative to positive when x increases through
The local minimum value of f (x) at x = \[\frac{3\pi}{4}\] is given by \[\sin\left( \frac{3\pi}{2} \right) = - 1\]
APPEARS IN
संबंधित प्रश्न
f(x) = - (x-1)2+2 on R ?
f(x)=sin 2x+5 on R .
f(x) = | sin 4x+3 | on R ?
f (x) = \[-\] | x + 1 | + 3 on R .
f(x) = 16x2 \[-\] 16x + 28 on R ?
f(x) = x3 \[-\] 1 on R .
f(x) = (x \[-\] 5)4.
f(x) = (x \[-\] 1) (x+2)2.
`f(x)=2sinx-x, -pi/2<=x<=pi/2`
Find the point of local maximum or local minimum, if any, of the following function, using the first derivative test. Also, find the local maximum or local minimum value, as the case may be:
f(x) = x3(2x \[-\] 1)3.
f(x) = x4 \[-\] 62x2 + 120x + 9.
f(x) = (x - 1) (x + 2)2.
`f(x) = (x+1) (x+2)^(1/3), x>=-2` .
`f(x)=xsqrt(32-x^2), -5<=x<=5` .
f(x) = \[x + \frac{a2}{x}, a > 0,\] , x ≠ 0 .
f(x) = \[x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}\] .
f(x) = \[x + \sqrt{1 - x}, x \leq 1\] .
f(x) = (x \[-\] 1) (x \[-\] 2)2.
f(x) = \[- (x - 1 )^3 (x + 1 )^2\] .
Find the maximum and minimum values of the function f(x) = \[\frac{4}{x + 2} + x .\]
Find the absolute maximum and minimum values of a function f given by `f(x) = 12 x^(4/3) - 6 x^(1/3) , x in [ - 1, 1]` .
A beam is supported at the two end and is uniformly loaded. The bending moment M at a distance x from one end is given by \[M = \frac{Wx}{3}x - \frac{W}{3}\frac{x^3}{L^2}\] .
Find the point at which M is maximum in a given case.
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 lengths of the two pieces so that the combined area of the circle and the square is minimum?
Given the sum of the perimeters of a square and a circle, show that the sum of there areas is least when one side of the square is equal to diameter of the circle.
A window in the form of a rectangle is surmounted by a semi-circular opening. The total perimeter of the window is 10 m. Find the dimension of the rectangular of the window to admit maximum light through the whole opening.
Show that the height of the cylinder of maximum volume that can be inscribed a sphere of radius R is \[\frac{2R}{\sqrt{3}} .\]
A closed cylinder has volume 2156 cm3. What will be the radius of its base so that its total surface area is minimum ?
Find the maximum slope of the curve y = \[- x^3 + 3 x^2 + 2x - 27 .\]
An open tank is to be constructed with a square base and vertical sides so as to contain a given quantity of water. Show that the expenses of lining with lead with be least, if depth is made half of width.
The total area of a page is 150 cm2. The combined width of the margin at the top and bottom is 3 cm and the side 2 cm. What must be the dimensions of the page in order that the area of the printed matter may be maximum?
A particle is moving in a straight line such that its distance at any time t is given by S = \[\frac{t^4}{4} - 2 t^3 + 4 t^2 - 7 .\] Find when its velocity is maximum and acceleration minimum.
Write the minimum value of f(x) = xx .
The minimum value of \[\frac{x}{\log_e x}\] is _____________ .
Let f(x) = x3+3x2 \[-\] 9x+2. Then, f(x) has _________________ .
If x+y=8, then the maximum value of xy is ____________ .
The minimum value of \[\left( x^2 + \frac{250}{x} \right)\] is __________ .
f(x) = 1+2 sin x+3 cos2x, `0<=x<=(2pi)/3` is ________________ .
A wire of length 34 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a rectangle whose length is twice its breadth. What should be the lengths of the two pieces, so that the combined area of the square and the rectangle is minimum?
Of all the closed right circular cylindrical cans of volume 128π cm3, find the dimensions of the can which has minimum surface area.
