Advertisements
Advertisements
प्रश्न
If x lies in the interval [0,1], then the least value of x2 + x + 1 is _______________ .
विकल्प
3
`3/4`
1
none of these
Advertisements
उत्तर
1
\[\text { Given: } f\left( x \right) = x^2 + x + 1\]
\[ \Rightarrow f'\left( x \right) = 2x + 1\]
\[\text { For a local maxima or a local minima, we must have } \]
\[f'\left( x \right) = 0\]
\[ \Rightarrow 2x + 1 = 0\]
\[ \Rightarrow 2x = - 1\]
\[ \Rightarrow x = \frac{- 1}{2} \not\in \left[ 0, 1 \right]\]
\[\text { At extreme points } : \]
\[ f\left( 0 \right) = 0\]
\[f\left( 1 \right) = 1 + 1 + 1 = 3 > 0\]
\[\text { So, x = 1 is a local minima }. \]
APPEARS IN
संबंधित प्रश्न
f(x) = 4x2 + 4 on R .
f(x)=sin 2x+5 on R .
f(x) = | sin 4x+3 | on R ?
f(x) = (x \[-\] 5)4.
f(x) = (x \[-\] 1) (x+2)2.
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) =\[\frac{x}{2} + \frac{2}{x} , x > 0\] .
f(x) = (x - 1) (x + 2)2.
`f(x) = x/2+2/x, x>0 `.
f(x) = \[- (x - 1 )^3 (x + 1 )^2\] .
Show that \[\frac{\log x}{x}\] has a maximum value at x = e ?
Find the maximum and minimum values of y = tan \[x - 2x\] .
f(x) = (x \[-\] 2) \[\sqrt{x - 1} \text { in }[1, 9]\] .
Find the absolute maximum and minimum values of a function f given by f(x) = 2x3 − 15x2 + 36x + 1 on the interval [1, 5].
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?
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 rectangle is inscribed in a semi-circle of radius r with one of its sides on diameter of semi-circle. Find the dimension of the rectangle so that its area is maximum. Find also the area ?
Find the point on the curve x2 = 8y which is nearest to the point (2, 4) ?
Find the coordinates of a point on the parabola y=x2+7x + 2 which is closest to the strainght line y = 3x \[-\] 3 ?
Find the point on the curvey y2 = 2x which is at a minimum distance from the point (1, 4).
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?
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 ?
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 .
The number which exceeds its square by the greatest possible quantity is _________________ .
The maximum value of f(x) = \[\frac{x}{4 - x + x^2}\] on [ \[-\] 1, 1] is _______________ .
Let x, y be two variables and x>0, xy=1, then minimum value of x+y is _______________ .
The function f(x) = \[2 x^3 - 15 x^2 + 36x + 4\] is maximum at x = ________________ .
The maximum value of f(x) = \[\frac{x}{4 + x + x^2}\] on [ \[-\] 1,1] 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.
Which of the following graph represents the extreme value:-
