Advertisements
Advertisements
प्रश्न
The least value of the function f(x) = \[x3 - 18x2 + 96x\] in the interval [0,9] is _____________ .
विकल्प
126
135
160
0
Advertisements
उत्तर
0
\[\text { Given }: f\left( x \right) = x^3 - 18 x^2 + 96x\]
\[ \Rightarrow f'\left( x \right) = 3 x^2 - 36x + 96\]
\[\text { For a local maxima or a local minima, we must have } \]
\[f'\left( x \right) = 0\]
\[ \Rightarrow 3 x^2 - 36x + 96 = 0\]
\[ \Rightarrow x^2 - 12x + 32 = 0\]
\[ \Rightarrow \left( x - 4 \right)\left( x - 8 \right) = 0\]
\[ \Rightarrow x = 4, 8\]
\[\text { So,} \]
\[f\left( 8 \right) = \left( 8 \right)^3 - 18 \left( 8 \right)^2 + 96\left( 8 \right) = 512 - 1152 + 768 = 128\]
\[f\left( 4 \right) = \left( 4 \right)^3 - 18 \left( 4 \right)^2 + 96\left( 4 \right) = 64 - 288 + 384 = 160\]
\[f\left( 0 \right) = \left( 0 \right)^3 - 18 \left( 0 \right)^2 + 96\left( 0 \right) = 0\]
\[f\left( 9 \right) = \left( 9 \right)^3 - 18 \left( 9 \right)^2 + 96\left( 9 \right) = 729 - 1458 + 864 = 135\]
\[\text { Hence, 0 is the minimum value in the range } \left[ 0, 9 \right] .\]
APPEARS IN
संबंधित प्रश्न
f(x) = 4x2 + 4 on R .
f(x) = | sin 4x+3 | on R ?
f(x) = x3 \[-\] 3x.
f(x) = x3 \[-\] 6x2 + 9x + 15 .
f(x) = sin x \[-\] cos x, 0 < x < 2\[\pi\] .
`f(x)=2sinx-x, -pi/2<=x<=pi/2`
f(x) =\[x\sqrt{1 - x} , x > 0\].
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) = 2/x - 2/x^2, x>0`
`f(x) = (x+1) (x+2)^(1/3), x>=-2` .
f(x) = \[x + \frac{a2}{x}, a > 0,\] , x ≠ 0 .
f(x) = \[- (x - 1 )^3 (x + 1 )^2\] .
Find the maximum and minimum values of the function f(x) = \[\frac{4}{x + 2} + x .\]
f(x) = 4x \[-\] \[\frac{x^2}{2}\] in [ \[-\] 2,4,5] .
Find the absolute maximum and minimum values of the function of given by \[f(x) = \cos^2 x + \sin x, x \in [0, \pi]\] .
Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.
Divide 15 into two parts such that the square of one multiplied with the cube of the other is minimum.
Of all the closed cylindrical cans (right circular), which enclose a given volume of 100 cm3, which has the minimum surface area?
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?
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.
Find the largest possible area of a right angled triangle whose hypotenuse is 5 cm long.
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 point on the parabolas x2 = 2y which is closest to the point (0,5) ?
Find the coordinates of a point on the parabola y=x2+7x + 2 which is closest to the strainght line y = 3x \[-\] 3 ?
A box of constant volume c is to be twice as long as it is wide. The material on the top and four sides cost three times as much per square metre as that in the bottom. What are the most economic dimensions?
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 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) = \[x + \frac{1}{x}, x > 0 .\]
The maximum value of x1/x, x > 0 is __________ .
The minimum value of \[\frac{x}{\log_e x}\] is _____________ .
Let f(x) = (x \[-\] a)2 + (x \[-\] b)2 + (x \[-\] c)2. Then, f(x) has a minimum at x = _____________ .
The maximum value of f(x) = \[\frac{x}{4 + x + x^2}\] on [ \[-\] 1,1] 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?
