Advertisements
Advertisements
प्रश्न
Find the point on the parabolas x2 = 2y which is closest to the point (0,5) ?
उत्तर
\[\text { Let the required point be } \left( x, y \right) . \text { Then }, \]
\[ x^2 = 2y\]
\[ \Rightarrow y = \frac{x^2}{2} .............\left( 1 \right)\]
\[\text { The distance between points } \left( x, y \right) \text { and } \left( 0, 5 \right) \text { is given by }\]
\[ d^2 = \left( x \right)^2 + \left( y - 5 \right)^2 \]
\[\text { Now,} \]
\[ d^2 = Z\]
\[ \Rightarrow Z = \left( x \right)^2 + \left( \frac{x^2}{2} - 5 \right)^2 \]
\[ \Rightarrow Z = x^2 + \frac{x^4}{4} + 25 - 5 x^2 \]
\[ \Rightarrow \frac{dZ}{dy} = 2x + x^3 - 10x\]
\[\text {For maximum or a minimum values of Z, we must have }\]
\[\frac{dZ}{dy} = 0\]
\[ \Rightarrow x^3 - 8x = 0\]
\[ \Rightarrow x^2 = 8\]
\[ \Rightarrow x = \pm 2\sqrt{2}\]
\[\text { Substituting the value of x in eq. }\left( 1 \right), \text { we get }\]
\[y = 4\]
\[\frac{d^2 Z}{d y^2} = 3 x^2 - 8\]
\[ \Rightarrow \frac{d^2 Z}{d y^2} = 24 - 8 = 16 > 0\]
\[\text { So, the nearest point is }\left( \pm 2\sqrt{2}, 4 \right) .\]
APPEARS IN
संबंधित प्रश्न
f(x) = (x \[-\] 1) (x+2)2.
f(x) = (x - 1) (x + 2)2.
`f(x) = 2/x - 2/x^2, x>0`
f(x) = xex.
f(x) = \[x^3 - 2a x^2 + a^2 x, a > 0, x \in R\] .
Show that \[\frac{\log x}{x}\] has a maximum value at x = e ?
Find the maximum and minimum values of the function f(x) = \[\frac{4}{x + 2} + x .\]
Find the maximum value of 2x3\[-\] 24x + 107 in the interval [1,3]. Find the maximum value of the same function in [ \[-\] 3, \[-\] 1].
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.
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 rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, in cutting off squares from each corners and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum possible?
An isosceles triangle of vertical angle 2 \[\theta\] is inscribed in a circle of radius a. Show that the area of the triangle is maximum when \[\theta\] = \[\frac{\pi}{6}\] .
Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible when revolved about one of its sides ?
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 \[5\sqrt{3 cm} \text { is }500 \pi {cm}^3 .\]
Determine the points on the curve x2 = 4y which are nearest to the point (0,5) ?
Find the point on the curvey y2 = 2x which is at a minimum distance from the point (1, 4).
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 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?
Write necessary condition for a point x = c to be an extreme point of the function f(x).
Write sufficient conditions for a point x = c to be a point of local maximum.
If f(x) attains a local minimum at x = c, then write the values of `f' (c)` and `f'' (c)`.
Write the point where f(x) = x log, x attains minimum value.
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 _________________ .
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 ______________ .
If x lies in the interval [0,1], then the least value of x2 + x + 1 is _______________ .
The maximum value of f(x) = \[\frac{x}{4 - x + x^2}\] on [ \[-\] 1, 1] is _______________ .
The point on the curve y2 = 4x which is nearest to, the point (2,1) is _______________ .
If x+y=8, then the maximum value of xy is ____________ .
The least and greatest values of f(x) = x3\[-\] 6x2+9x in [0,6], are ___________ .
If(x) = x+\[\frac{1}{x}\],x > 0, then its greatest value is _______________ .
Which of the following graph represents the extreme value:-
