Advertisements
Advertisements
प्रश्न
f(x) = \[x + \sqrt{1 - x}, x \leq 1\] .
उत्तर
\[\text { Given }: f\left( x \right) = x + \sqrt{1 - x}\]
\[ \Rightarrow f'\left( x \right) = 1 - \frac{1}{2\sqrt{1 - x}}\]
\[\text { For the local maxima or minima, we must have }\]
\[ f'\left( x \right) = 0\]
\[ \Rightarrow 1 - \frac{1}{2\sqrt{1 - x}} = 0\]
\[ \Rightarrow \sqrt{1 - x} = \frac{1}{2}\]
\[ \Rightarrow 1 - x = \frac{1}{4} \]
\[ \Rightarrow x = \frac{3}{4} \]
\[\text { Thus }, x = \frac{3}{4} \text { is the possible point of local maxima or local minima }. \]
\[\text { Now }, \]
\[f''\left( x \right) = - \frac{\frac{1}{4\sqrt{1 - x}}}{4\left( 1 - x \right)}\]
\[\text { At }x = \frac{3}{4}: \]
\[ f''\left( \frac{3}{4} \right) = - \frac{\frac{1}{4\sqrt{1 - \frac{3}{4}}}}{4\left( 1 - \frac{3}{4} \right)} = - \frac{1}{2} < 0\]
\[\text { So,} x = \frac{3}{4} \text { is the point of local maximum }. \]
\[\text { The local maximum value is given by }\]
\[f\left( \frac{3}{4} \right) = \frac{3}{4} + \sqrt{1 - \frac{3}{4}} = \frac{5}{4}\]
APPEARS IN
संबंधित प्रश्न
f(x) = 4x2 + 4 on R .
f (x) = \[-\] | x + 1 | + 3 on R .
f(x) = sin x \[-\] cos x, 0 < x < 2\[\pi\] .
`f(x)=sin2x-x, -pi/2<=x<=pi/2`
`f(x)=2sinx-x, -pi/2<=x<=pi/2`
f(x) =\[\frac{x}{2} + \frac{2}{x} , x > 0\] .
`f(x) = x/2+2/x, x>0 `.
`f(x)=xsqrt(32-x^2), -5<=x<=5` .
f(x) = (x \[-\] 1) (x \[-\] 2)2.
Show that \[\frac{\log x}{x}\] has a maximum value at x = e ?
If f(x) = x3 + ax2 + bx + c has a maximum at x = \[-\] 1 and minimum at x = 3. Determine a, b and c ?
`f(x) = 3x^4 - 8x^3 + 12x^2- 48x + 25 " in "[0,3]` .
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 \[f(x) = 2 x^3 - 15 x^2 + 36x + 1 \text { on the interval } [1, 5]\] ?
Divide 64 into two parts such that the sum of the cubes of two parts 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?
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 the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm ?
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) ?
Find the maximum slope of the curve y = \[- x^3 + 3 x^2 + 2x - 27 .\]
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 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?
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) = \[x + \frac{1}{x}, x > 0 .\]
The minimum value of f(x) = \[x4 - x2 - 2x + 6\] is _____________ .
The number which exceeds its square by the greatest possible quantity 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 ______________ .
The function f(x) = \[\sum^5_{r = 1}\] (x \[-\] r)2 assumes minimum value at x = ______________ .
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 ______________ .
The minimum value of the function `f(x)=2x^3-21x^2+36x-20` is ______________ .
Which of the following graph represents the extreme value:-
