Advertisements
Advertisements
प्रश्न
Find the point on the curve x2 = 8y which is nearest to the point (2, 4) ?
Advertisements
उत्तर
\[\text { Let }\left( x, y \right) \text {be nearest to the point } \left( 2, 4 \right) . \text { Then }, \]
\[ x^2 = 8y\]
\[ \Rightarrow y = \frac{x^2}{8} ............\left( 1 \right)\]
\[ d^2 = \left( x - 2 \right)^2 + \left( y - 4 \right)^2 ................\left[ \text {Using distance formula} \right]\]
\[\text { Now,} \]
\[Z = d^2 = \left( x - 2 \right)^2 + \left( y - 4 \right)^2 \]
\[ \Rightarrow Z = \left( x - 2 \right)^2 + \left( \frac{x^2}{8} - 4 \right)^2 .............\left[\text {From eq. } \left( 1 \right) \right]\]
\[ \Rightarrow Z = x^2 + 4 - 4x + \frac{x^4}{64} + 16 - x^2 \]
\[ \Rightarrow \frac{dZ}{dy} = - 4 + \frac{4 x^3}{64}\]
\[\text {For maximum or minimum values of Z, we must have }\]
\[\frac{dZ}{dy} = 0\]
\[ \Rightarrow - 4 + \frac{4 x^3}{64} = 0\]
\[ \Rightarrow \frac{x^3}{16} = 4\]
\[ \Rightarrow x^3 = 64\]
\[ \Rightarrow x = 4\]
\[\text { Substituting the value of x in eq. } \left( 1 \right), \text { we get }\]
\[y = 2\]
\[\text { Now,} \]
\[\frac{d^2 Z}{d y^2} = \frac{12 x^2}{64}\]
\[ \Rightarrow \frac{d^2 Z}{d y^2} = 3 > 0\]
\[\text { So, the nearest point is } \left( 4, 2 \right) .\]
APPEARS IN
संबंधित प्रश्न
f(x) = 4x2 + 4 on R .
f(x) = x3 (x \[-\] 1)2 .
f(x) = \[\frac{1}{x^2 + 2}\] .
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) = x4 \[-\] 62x2 + 120x + 9.
f(x) = (x - 1) (x + 2)2.
`f(x) = (x+1) (x+2)^(1/3), x>=-2` .
`f(x)=xsqrt(32-x^2), -5<=x<=5` .
f(x) = \[x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}\] .
f(x) = \[x + \sqrt{1 - x}, x \leq 1\] .
f(x) = \[- (x - 1 )^3 (x + 1 )^2\] .
f(x) = (x \[-\] 2) \[\sqrt{x - 1} \text { in }[1, 9]\] .
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?
Of all the closed cylindrical cans (right circular), which enclose a given volume of 100 cm3, which has the minimum surface area?
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 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 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?
Show that the height of the cylinder of maximum volume that can be inscribed a sphere of radius R is \[\frac{2R}{\sqrt{3}} .\]
Prove that a conical tent of given capacity will require the least amount of canavas when the height is \[\sqrt{2}\] times the radius of the base.
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) ?
The strength of a beam varies as the product of its breadth and square of its depth. Find the dimensions of the strongest beam which can be cut from a circular log of radius a ?
If f(x) attains a local minimum at x = c, then write the values of `f' (c)` and `f'' (c)`.
Write the minimum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
Write the minimum value of f(x) = xx .
Write the maximum value of f(x) = \[\frac{\log x}{x}\], if it exists .
The sum of two non-zero numbers is 8, the minimum value of the sum of the reciprocals is ______________ .
The least value of the function f(x) = \[x3 - 18x2 + 96x\] in the interval [0,9] 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 maximum value of f(x) = \[\frac{x}{4 + x + x^2}\] on [ \[-\] 1,1] is ___________________ .
Of all the closed right circular cylindrical cans of volume 128π cm3, find the dimensions of the can which has minimum surface area.
Which of the following graph represents the extreme value:-
