Advertisements
Advertisements
प्रश्न
The least and greatest values of f(x) = x3\[-\] 6x2+9x in [0,6], are ___________ .
विकल्प
3, 4
0, 6
0, 3
3, 6
0, 54
उत्तर
\[\text { Given: } f\left( x \right) = x^3 - 6 x^2 + 9x\]
\[ \Rightarrow f'\left( x \right) = 3 x^2 - 12x + 9\]
\[\text { For a local maxima or a local minima, we must have }\]
\[f'\left( x \right) = 0\]
\[ \Rightarrow 3 x^2 - 12x + 9 = 0\]
\[ \Rightarrow x^2 - 4x + 3 = 0\]
\[ \Rightarrow \left( x - 1 \right)\left( x - 3 \right) = 0\]
\[ \Rightarrow x = 1, 3\]
\[\text { Now,} \]
\[f\left( 0 \right) = 0^3 - 6 \left( 0 \right)^2 + 9\left( 0 \right) = 0\]
\[f\left( 1 \right) = 1^3 - 6 \left( 1 \right)^2 + 9\left( 1 \right) = 1 - 6 + 9 = 4\]
\[f\left( 3 \right) = 3^3 - 6 \left( 3 \right)^2 + 9\left( 3 \right) = 27 - 54 + 27 = 0\]
\[f\left( 6 \right) = 6^3 - 6 \left( 6 \right)^2 + 9\left( 6 \right) = 216 - 216 + 54 = 54\]
The least and greatest values of f(x) = x3- 6x2+9x in [0, 6] are 0 and 54, respectively.
Notes
The question in the book has some error. So, none of the options are matching with the solution. The solution here is according to the question given in the book.
APPEARS IN
संबंधित प्रश्न
f(x)=| x+2 | on R .
f(x) = x3 \[-\] 1 on R .
f(x) = x3 \[-\] 3x .
f(x) = (x \[-\] 1) (x+2)2.
f(x) = sin x \[-\] cos x, 0 < x < 2\[\pi\] .
f(x) =\[x\sqrt{1 - x} , x > 0\].
f(x) = x4 \[-\] 62x2 + 120x + 9.
`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)=xsqrt(1-x), x<=1` .
If f(x) = x3 + ax2 + bx + c has a maximum at x = \[-\] 1 and minimum at x = 3. Determine a, b and c ?
Prove that f(x) = sinx + \[\sqrt{3}\] cosx has maximum value at x = \[\frac{\pi}{6}\] ?
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 \[M = \frac{Wx}{3}x - \frac{W}{3}\frac{x^3}{L^2}\] .
Find the point at which M is maximum in a given case.
A wire of length 20 m is to be cut into two pieces. One of the pieces will be bent into shape of a square and the other into shape of an equilateral triangle. Where the we should be cut so that the sum of the areas of the square and triangle 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.
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is \[6\sqrt{3}\]r.
A closed cylinder has volume 2156 cm3. What will be the radius of its base so that its total surface area is minimum ?
Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius \[5\sqrt{3 cm} \text { is }500 \pi {cm}^3 .\]
The total cost of producing x radio sets per day is Rs \[\left( \frac{x^2}{4} + 35x + 25 \right)\] and the price per set at which they may be sold is Rs. \[\left( 50 - \frac{x}{2} \right) .\] Find the daily output to maximum the total profit.
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.
Write the minimum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
Find the least value of f(x) = \[ax + \frac{b}{x}\], where a > 0, b > 0 and x > 0 .
Write the maximum value of f(x) = x1/x.
Write the maximum value of f(x) = \[\frac{\log x}{x}\], if it exists .
For the function f(x) = \[x + \frac{1}{x}\]
The number which exceeds its square by the greatest possible quantity is _________________ .
The sum of two non-zero numbers is 8, the minimum value of the sum of the reciprocals is ______________ .
The function f(x) = \[\sum^5_{r = 1}\] (x \[-\] r)2 assumes minimum value at x = ______________ .
The maximum value of f(x) = \[\frac{x}{4 - x + x^2}\] on [ \[-\] 1, 1] is _______________ .
If x+y=8, then the maximum value of xy is ____________ .
The minimum value of \[\left( x^2 + \frac{250}{x} \right)\] is __________ .
The maximum value of f(x) = \[\frac{x}{4 + x + x^2}\] on [ \[-\] 1,1] is ___________________ .
The minimum value of x loge x is equal to ____________ .
The minimum value of the function `f(x)=2x^3-21x^2+36x-20` is ______________ .
