Advertisements
Advertisements
प्रश्न
f(x) = x3 \[-\] 3x.
Advertisements
उत्तर
\[\text { Given: } \hspace{0.167em} f\left( x \right) = \left( x - 5 \right)^4 \]
\[ \Rightarrow f'\left( x \right) = 4 \left( x - 5 \right)^3 \]
\[\text { For a local maximum or a local minimum, we must have }\]
\[f'\left( x \right) = 0\]
\[ \Rightarrow 4 \left(x - 5 \right)^3 = 0\]
\[ \Rightarrow x = 5\]
Since f '(x) changes from negative to positive as x increases through 1, x = 1 is the point of local minima.
The local minimum value of f (x) at x = 1 is given by \[\left( 1 \right)^3 - 3\left( 1 \right) = - 2\]
Since f '(x) changes from positive to negative when x increases through -1, x = -1 is the point of local maxima.
The local maximum value of f (x) at x = -1 is given by \[\left( - 1 \right)^3 - 3\left( - 1 \right) = 2\]
APPEARS IN
संबंधित प्रश्न
f(x) = 4x2 + 4 on R .
f(x) = - (x-1)2+2 on R ?
f(x)=| x+2 | on R .
f(x) = x3 (x \[-\] 1)2 .
f(x) = (x \[-\] 1) (x+2)2.
f(x) = cos x, 0 < x < \[\pi\] .
`f(x)=sin2x-x, -pi/2<=x<=pi/2`
f(x) = \[- (x - 1 )^3 (x + 1 )^2\] .
Show that \[\frac{\log x}{x}\] has a maximum value at x = e ?
If f(x) = x3 + ax2 + bx + c has a maximum at x = \[-\] 1 and minimum at x = 3. Determine a, b and c ?
Find the maximum value of 2x3\[-\] 24x + 107 in the interval [1,3]. Find the maximum value of the same function in [ \[-\] 3, \[-\] 1].
How should we choose two numbers, each greater than or equal to `-2, `whose sum______________ so that the sum of the first and the cube of the second is minimum?
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 \[M = \frac{WL}{2}x - \frac{W}{2} x^2\] .
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?
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 a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Find this maximum volume.
A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, in cutting off squares from each corners 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 possible?
Show that the height of the cylinder of maximum volume that can be inscribed a sphere of radius R is \[\frac{2R}{\sqrt{3}} .\]
Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm ?
A closed cylinder has volume 2156 cm3. What will be the radius of its base so that its total surface area is minimum ?
Determine the points on the curve x2 = 4y which are nearest to the point (0,5) ?
Find the point on the curve x2 = 8y which is nearest to the point (2, 4) ?
Find the maximum slope of the curve y = \[- x^3 + 3 x^2 + 2x - 27 .\]
Manufacturer can sell x items at a price of rupees \[\left( 5 - \frac{x}{100} \right)\] each. The cost price is Rs \[\left( \frac{x}{5} + 500 \right) .\] Find the number of items he should sell to earn maximum profit.
The sum of the surface areas of a sphere and a cube is given. Show that when the sum of their volumes is least, the diameter of the sphere is equal to the edge of the cube.
The space s described in time t by a particle moving in a straight line is given by S = \[t5 - 40 t^3 + 30 t^2 + 80t - 250 .\] Find the minimum value of acceleration.
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.
If f(x) attains a local minimum at x = c, then write the values of `f' (c)` and `f'' (c)`.
Write the maximum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
Write the point where f(x) = x log, x attains minimum value.
Write the maximum value of f(x) = \[\frac{\log x}{x}\], if it exists .
The minimum value of f(x) = \[x4 - x2 - 2x + 6\] is _____________ .
Let f(x) = (x \[-\] a)2 + (x \[-\] b)2 + (x \[-\] c)2. Then, f(x) has a minimum at x = _____________ .
If x+y=8, then the maximum value of xy is ____________ .
The minimum value of \[\left( x^2 + \frac{250}{x} \right)\] is __________ .
If(x) = \[\frac{1}{4x^2 + 2x + 1}\] then its maximum value is _________________ .
f(x) = 1+2 sin x+3 cos2x, `0<=x<=(2pi)/3` is ________________ .
