Advertisements
Advertisements
प्रश्न
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 ?
Advertisements
उत्तर
\[\text { The equation of line passing through }\left( 1, 4 \right) \text { with slope m is given by } \]
\[y - 4 = m\left( x - 1 \right) ................. \left( 1 \right)\]
\[\text { Substituting y = 0, we get }\]
\[0 - 4 = m\left( x - 1 \right)\]
\[ \Rightarrow \frac{- 4}{m} = x - 1\]
\[ \Rightarrow x = \frac{m - 4}{m}\]
\[\text{ Substituting x = 0, we get } \]
\[y - 4 = m\left( 0 - 1 \right)\]
\[ \Rightarrow y = - m + 4\]
\[ \Rightarrow x = - \left( m - 4 \right)\]
\[\text { So, the intercepts on coordinate axes are } \frac{m - 4}{m} \text { and }- \left( m - 4 \right) . \]
\[\text { Let S be the sum of the intercepts . Then }, \]
\[S = \frac{m - 4}{m} - \left( m - 4 \right)\]
\[ \Rightarrow \frac{dS}{dm} = \frac{4}{m^2} - 1\]
\[\text { For maximum or minimum values of S, we must have }\]
\[ \frac{dS}{dm} = 0\]
\[ \Rightarrow \frac{4}{m^2} - 1 = 0\]
\[ \Rightarrow \frac{4}{m^2} = 1\]
\[ \Rightarrow m^2 = 4\]
\[ \Rightarrow m = \pm 2\]
\[\text {Now}, \]
\[\frac{d^2 S}{d m^2} = \frac{- 8}{m^3}\]
\[ \left( \frac{d^2 S}{d m^2} \right)_{m = 2} = \frac{- 8}{2^3} = - 1 < 0\]
\[\text { So, the sum is minimum at m = 2} . \]
\[ \left( \frac{d^2 S}{d m^2} \right)_{m = - 2} = \frac{- 8}{\left( - 2 \right)^3} = 1 > 0\]
\[S\text { o, the sum is maximum at m = - 2 } . \]
\[\text { Thus, the minimum value is given by}\]
\[S = \frac{- 2 - 4}{- 2} - \left( - 2 - 4 \right) = 3 + 6 = 9\]
APPEARS IN
संबंधित प्रश्न
f(x) = | sin 4x+3 | on R ?
f(x)=2x3 +5 on R .
f(x) = 16x2 \[-\] 16x + 28 on R ?
f(x) = x3 \[-\] 1 on R .
f(x) =\[x\sqrt{1 - x} , x > 0\].
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 - 1) (x + 2)2.
`f(x) = x/2+2/x, x>0 `.
f(x) = \[x^3 - 2a x^2 + a^2 x, a > 0, x \in R\] .
f(x) = \[x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}\] .
f(x) = \[x + \sqrt{1 - x}, x \leq 1\] .
Find the absolute maximum and minimum values of a function f given by f(x) = 2x3 − 15x2 + 36x + 1 on the interval [1, 5].
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 \[M = \frac{WL}{2}x - \frac{W}{2} x^2\] .
Find the point at which M is maximum in a given case.
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?
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 x2 = 8y which is nearest to the point (2, 4) ?
Find the point on the parabolas x2 = 2y which is closest to the point (0,5) ?
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.
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 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 ?
A particle is moving in a straight line such that its distance at any time t is given by S = \[\frac{t^4}{4} - 2 t^3 + 4 t^2 - 7 .\] Find when its velocity is maximum and acceleration minimum.
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.
Find the least value of f(x) = \[ax + \frac{b}{x}\], where a > 0, b > 0 and x > 0 .
For the function f(x) = \[x + \frac{1}{x}\]
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 function f(x) = \[2 x^3 - 15 x^2 + 36x + 4\] is maximum at x = ________________ .
The minimum value of x loge x is equal to ____________ .
