Advertisements
Advertisements
प्रश्न
The number which exceeds its square by the greatest possible quantity is _________________ .
पर्याय
\[\frac{1}{2}\]
\[\frac{1}{4}\]
\[\frac{3}{4}\]
none of these
Advertisements
उत्तर
\[\frac{1}{2}\]
\[\text { Let the required number be x . Then, } \]
\[f\left( x \right) = x - x^2 \]
\[ \Rightarrow f'\left( x \right) = 1 - 2x\]
\[\text { For a local maxima or a local minima, we must have } \]
\[f'\left( x \right) = 0\]
\[ \Rightarrow 1 - 2x = 0\]
\[ \Rightarrow 2x = 1\]
\[ \Rightarrow x = \frac{1}{2}\]
\[\text { Now }, \]
\[f''\left( x \right) = - 2 < 0\]
\[\text { So, } x = \frac{1}{2}\text { is a local maxima }. \]
\[\text { Hence, the required number is } \frac{1}{2} . \]
APPEARS IN
संबंधित प्रश्न
f(x)=| x+2 | on R .
f(x) = | sin 4x+3 | on R ?
f(x)=2x3 +5 on R .
f(x) = x3 \[-\] 1 on R .
f(x) = x3 (x \[-\] 1)2 .
f(x) = (x \[-\] 1) (x+2)2.
f(x) = \[\frac{1}{x^2 + 2}\] .
f(x) = x4 \[-\] 62x2 + 120x + 9.
`f(x) = 2/x - 2/x^2, x>0`
`f(x) = x/2+2/x, x>0 `.
`f(x)=xsqrt(32-x^2), -5<=x<=5` .
f(x) = (x \[-\] 1) (x \[-\] 2)2.
Prove that f(x) = sinx + \[\sqrt{3}\] cosx has maximum value at x = \[\frac{\pi}{6}\] ?
Find the maximum value of 2x3\[-\] 24x + 107 in the interval [1,3]. Find the maximum value of the same function in [ \[-\] 3, \[-\] 1].
Find the absolute maximum and minimum values of the function of given by \[f(x) = \cos^2 x + \sin x, x \in [0, \pi]\] .
Find the absolute maximum and minimum values of a function f given by `f(x) = 12 x^(4/3) - 6 x^(1/3) , x in [ - 1, 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?
Divide 15 into two parts such that the square of one multiplied with the cube of the other is minimum.
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 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?
Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm ?
Find the point on the parabolas x2 = 2y which is closest to the point (0,5) ?
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 ?
Write necessary condition for a point x = c to be an extreme point of the function f(x).
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 .
The minimum value of \[\frac{x}{\log_e x}\] is _____________ .
For the function f(x) = \[x + \frac{1}{x}\]
Let f(x) = x3+3x2 \[-\] 9x+2. Then, f(x) has _________________ .
If x lies in the interval [0,1], then the least value of x2 + x + 1 is _______________ .
The least value of the function f(x) = \[x3 - 18x2 + 96x\] in the interval [0,9] is _____________ .
The least and greatest values of f(x) = x3\[-\] 6x2+9x in [0,6], are ___________ .
The minimum value of \[\left( x^2 + \frac{250}{x} \right)\] is __________ .
If(x) = x+\[\frac{1}{x}\],x > 0, then its greatest value 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.
