Advertisements
Advertisements
प्रश्न
Let f(x) = x3+3x2 \[-\] 9x+2. Then, f(x) has _________________ .
विकल्प
a maximum at x = 1
a minimum at x = 1
neither a maximum nor a minimum at x = - 3
none of these
Advertisements
उत्तर
\[\text { a minimum at x = 1}\]
\[\text { Given }: f\left( x \right) = x^3 + 3 x^2 - 9x + 2\]
\[ \Rightarrow f'\left( x \right) = 3 x^2 + 6x - 9\]
\[\text { For a local maxima or a local minima, we must have } \]
\[f'\left( x \right) = 0\]
\[ \Rightarrow 3 x^2 + 6x - 9 = 0\]
\[ \Rightarrow x^2 + 2x - 3 = 0\]
\[ \Rightarrow \left( x + 3 \right)\left( x - 1 \right) = 0\]
\[ \Rightarrow x = - 3, 1\]
\[\text { Now,} \]
\[f''\left( x \right) = 6x + 6\]
\[ \Rightarrow f''\left( 1 \right) = 6 + 6 = 12 > 0\]
\[\text { So, x = 1 is a local minima } . \]
\[\text { Also }, \]
\[f''\left( - 3 \right) = - 18 + 6 = - 12 < 0\]
\[\text { So, x = - 3 is a local maxima } . \]
APPEARS IN
संबंधित प्रश्न
f(x) = x3 \[-\] 1 on R .
f(x) = x3 \[-\] 3x.
f(x) = x3 (x \[-\] 1)2 .
f(x) = (x \[-\] 1) (x+2)2.
f(x) = \[\frac{1}{x^2 + 2}\] .
f(x) = cos x, 0 < x < \[\pi\] .
`f(x)=2sinx-x, -pi/2<=x<=pi/2`
f(x) =\[\frac{x}{2} + \frac{2}{x} , x > 0\] .
f(x) = x4 \[-\] 62x2 + 120x + 9.
`f(x) = 2/x - 2/x^2, x>0`
f(x) = xex.
`f(x)=xsqrt(32-x^2), -5<=x<=5` .
f(x) = \[x + \frac{a2}{x}, a > 0,\] , x ≠ 0 .
f(x) = (x \[-\] 1) (x \[-\] 2)2.
`f(x)=xsqrt(1-x), x<=1` .
Find the maximum and minimum values of the function f(x) = \[\frac{4}{x + 2} + x .\]
Prove that f(x) = sinx + \[\sqrt{3}\] cosx has maximum value at x = \[\frac{\pi}{6}\] ?
f(x) = (x \[-\] 1)2 + 3 in [ \[-\] 3,1] ?
`f(x) = 3x^4 - 8x^3 + 12x^2- 48x + 25 " in "[0,3]` .
Find the maximum value of 2x3\[-\] 24x + 107 in the interval [1,3]. Find the maximum value of the same function in [ \[-\] 3, \[-\] 1].
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?
Two sides of a triangle have lengths 'a' and 'b' and the angle between them is \[\theta\]. What value of \[\theta\] will maximize the area of the triangle? Find the maximum area of the triangle also.
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 tank with rectangular base and rectangular sides, open at the top, is to the constructed so that its depth is 2 m and volume is 8 m3. If building of tank cost 70 per square metre for the base and Rs 45 per square metre for sides, what is the cost of least expensive tank?
A large window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 metres find the dimensions of the rectangle will produce the largest area of the window.
Find the maximum slope of the curve y = \[- x^3 + 3 x^2 + 2x - 27 .\]
A straight line is drawn through a given point P(1,4). Determine the least value of the sum of the intercepts on the coordinate axes ?
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) = \[x + \frac{1}{x}, x > 0 .\]
Write the point where f(x) = x log, x attains minimum value.
The number which exceeds its square by the greatest possible quantity is _________________ .
Let f(x) = (x \[-\] a)2 + (x \[-\] b)2 + (x \[-\] c)2. Then, f(x) has a minimum at x = _____________ .
The sum of two non-zero numbers is 8, the minimum value of the sum of the reciprocals is ______________ .
If a cone of maximum volume is inscribed in a given sphere, then the ratio of the height of the cone to the diameter of the sphere 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 ________________ .
The sum of the surface areas of a cuboid with sides x, 2x and \[\frac{x}{3}\] and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of sphere. Also find the minimum value of the sum of their volumes.
Of all the closed right circular cylindrical cans of volume 128π cm3, find the dimensions of the can which has minimum surface area.
