Advertisements
Advertisements
प्रश्न
Find the coordinates of a point on the parabola y=x2+7x + 2 which is closest to the strainght line y = 3x \[-\] 3 ?
उत्तर
\[\text { Let coordinates of the point on the parabola be } \left( x, y \right) . \text { Then }, \]
\[y = x^2 + 7x + 2 ............. \left( 1 \right)\]
\[\text { Let the distance of a point } \left( x, \left( x^2 + 7x + 2 \right) \right) \text { from the line } y = 3x - 3\text { be S . Then,} \]
\[S = \left| \frac{- 3x + \left( x^2 + 7x + 2 \right) + 3}{\sqrt{10}} \right|\]
\[ \Rightarrow \frac{dS}{dt} = \frac{- 3 + 2x + 7}{\sqrt{10}}\]
\[\text { For maximum or minimum values of S, we must have }\]
\[\frac{dS}{dt} = 0\]
\[ \Rightarrow \frac{- 3 + 2x + 7}{\sqrt{10}} = 0\]
\[ \Rightarrow 2x = - 4\]
\[ \Rightarrow x = - 2\]
\[\text { Now }, \]
\[\frac{d^2 S}{d t^2} = \frac{2}{\sqrt{10}} > 0\]
\[\text { So, the nearest point is} \left( x, \left( x^2 + 7x + 2 \right) \right) . \]
\[ \Rightarrow \left( - 2, 4 - 14 + 2 \right)\]
\[ \Rightarrow \left( - 2, - 8 \right)\]
APPEARS IN
संबंधित प्रश्न
f(x)=sin 2x+5 on R .
f(x) = | sin 4x+3 | on R ?
f (x) = \[-\] | x + 1 | + 3 on R .
f(x) = x3 \[-\] 3x .
f(x) = \[\frac{1}{x^2 + 2}\] .
f(x) = sin x \[-\] cos x, 0 < x < 2\[\pi\] .
`f(x)=2sinx-x, -pi/2<=x<=pi/2`
f(x) =\[x\sqrt{1 - x} , x > 0\].
f(x) = xex.
f(x) = \[x + \frac{a2}{x}, a > 0,\] , x ≠ 0 .
f(x) = \[- (x - 1 )^3 (x + 1 )^2\] .
Show that \[\frac{\log x}{x}\] has a maximum value at x = e ?
Prove that f(x) = sinx + \[\sqrt{3}\] cosx has maximum value at x = \[\frac{\pi}{6}\] ?
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]` .
Find the absolute maximum and minimum values of a function f given by \[f(x) = 2 x^3 - 15 x^2 + 36x + 1 \text { on the interval } [1, 5]\] ?
Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.
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?
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 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 maximum volume of the cylinder which can be inscribed in a sphere of radius \[5\sqrt{3 cm} \text { is }500 \pi {cm}^3 .\]
Find the point on the curve y2 = 4x which is nearest to the point (2,\[-\] 8).
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?
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?
The space s described in time t by a particle moving in a straight line is given by S = \[t5 - 40 t^3 + 30 t^2 + 80t - 250 .\] Find the minimum value of acceleration.
Write the point where f(x) = x log, x attains minimum value.
Write the minimum value of f(x) = xx .
The maximum value of x1/x, x > 0 is __________ .
Let f(x) = (x \[-\] a)2 + (x \[-\] b)2 + (x \[-\] c)2. Then, f(x) has a minimum at x = _____________ .
The sum of two non-zero numbers is 8, the minimum value of the sum of the reciprocals is ______________ .
At x= \[\frac{5\pi}{6}\] f(x) = 2 sin 3x + 3 cos 3x is ______________ .
The least value of the function f(x) = \[x3 - 18x2 + 96x\] in the interval [0,9] is _____________ .
The point on the curve y2 = 4x which is nearest to, the point (2,1) is _______________ .
The least and greatest values of f(x) = x3\[-\] 6x2+9x in [0,6], are ___________ .
f(x) = \[\sin + \sqrt{3} \cos x\] is maximum when x = ___________ .
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.
Which of the following graph represents the extreme value:-
