Advertisements
Advertisements
प्रश्न
Find the point on the curve y2 = 4x which is nearest to the point (2,\[-\] 8).
उत्तर
\[\text {Let point }\left( x, y \right)\text { be the nearest to the point } \left( 2, - 8 \right) . \text { Then }, \]
\[ y^2 = 4x\]
\[ \Rightarrow x = \frac{y^2}{4} . . . \left( 1 \right)\]
\[ d^2 = \left( x - 2 \right)^2 + \left( y + 8 \right)^2 ............\left[ \text {Using distance formula } \right]\]
\[\text { Now,} \]
\[Z = d^2 = \left( x - 2 \right)^2 + \left( y + 8 \right)^2 \]
\[ \Rightarrow Z = \left( \frac{y^2}{4} - 2 \right)^2 + \left( y + 8 \right)^2 ................\left[ \text { From eq. }\left( 1 \right) \right]\]
\[ \Rightarrow Z = \frac{y^4}{16} + 4 - y^2 + y^2 + 64 + 16y\]
\[ \Rightarrow \frac{dZ}{dy} = \frac{4 y^3}{16} + 16\]
\[\text { For maximum or minimum values of Z, we must have }\]
\[\frac{dZ}{dy} = 0\]
\[ \Rightarrow \frac{4 y^3}{16} + 16 = 0\]
\[ \Rightarrow \frac{4 y^3}{16} = - 16\]
\[ \Rightarrow y^3 = - 64\]
\[ \Rightarrow y = - 4\]
\[\text { Substituting the value of x in eq. } \left( 1 \right), \text { we get }\]
\[x = 4\]
\[\text { Now, }\]
\[\frac{d^2 Z}{d y^2} = \frac{12 y^2}{16}\]
\[ \Rightarrow \frac{d^2 Z}{d y^2} = 12 > 0\]
\[\text { So, the nearest point is }\left( 4, - 4 \right) .\]
APPEARS IN
संबंधित प्रश्न
f(x)=sin 2x+5 on R .
f(x)=2x3 +5 on R .
f(x) = x3 (x \[-\] 1)2 .
f(x) = \[\frac{1}{x^2 + 2}\] .
f(x) = sin x \[-\] cos x, 0 < x < 2\[\pi\] .
`f(x)=sin2x-x, -pi/2<=x<=pi/2`
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) = x/2+2/x, x>0 `.
f(x) = \[x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}\] .
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 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 64 into two parts such that the sum of the cubes of two parts is minimum.
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.
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.
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.
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.
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is \[6\sqrt{3}\]r.
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 ?
Show that among all positive numbers x and y with x2 + y2 =r2, the sum x+y is largest when x=y=r \[\sqrt{2}\] .
Determine the points on the curve x2 = 4y which are nearest to the point (0,5) ?
Find the point on the parabolas x2 = 2y which is closest 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.
Manufacturer can sell x items at a price of rupees \[\left( 5 - \frac{x}{100} \right)\] each. The cost price is Rs \[\left( \frac{x}{5} + 500 \right) .\] Find the number of items he should sell to earn maximum profit.
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.
Write the maximum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
Write the point where f(x) = x log, x attains minimum value.
For the function f(x) = \[x + \frac{1}{x}\]
Let f(x) = x3+3x2 \[-\] 9x+2. Then, f(x) has _________________ .
If a cone of maximum volume is inscribed in a given sphere, then the ratio of the height of the cone to the diameter of the sphere is ______________ .
Let x, y be two variables and x>0, xy=1, then minimum value of x+y is _______________ .
The maximum value of f(x) = \[\frac{x}{4 + x + x^2}\] on [ \[-\] 1,1] is ___________________ .
Let f(x) = 2x3\[-\] 3x2\[-\] 12x + 5 on [ 2, 4]. The relative maximum occurs at x = ______________ .
The minimum value of x loge x is equal to ____________ .
Which of the following graph represents the extreme value:-
