Advertisements
Advertisements
Question
If(x) = \[\frac{1}{4x^2 + 2x + 1}\] then its maximum value is _________________ .
Options
\[\frac{4}{3}\]
\[\frac{2}{3}\]
1
\[\frac{3}{4}\]
Advertisements
Solution
\[\frac{4}{3}\]
\[\text { Maximum value of }\frac{1}{4 x^2 + 2x + 1}= \text { Minimum value of }4 x^2 + 2x + 1 \]
\[\text{ Now, }\]
\[f\left( x \right) = 4 x^2 + 2x + 1\]
\[ \Rightarrow f'\left( x \right) = 8x + 2\]
\[\text { For a local maxima or a local minima, we must have } \]
\[f'\left( x \right) = 0\]
\[ \Rightarrow 8x + 2 = 0\]
\[ \Rightarrow 8x = - 2\]
\[ \Rightarrow x = \frac{- 1}{4}\]
\[\text { Now, } \]
\[f''\left( x \right) = 8\]
\[ \Rightarrow f''\left( 1 \right) = 8 > 0\]
\[\text { So, x } = \frac{- 1}{4} \text { is a local minima } . \]
\[\text { Thus },\frac{1}{4 x^2 + 2x + 1}\text { is maximum at x } = \frac{- 1}{4} . \]
\[ \Rightarrow \text { Maximum value of } \frac{1}{4 x^2 + 2x + 1} = \frac{1}{4 \left( \frac{- 1}{4} \right)^2 + 2\left( \frac{- 1}{4} \right) + 1}\]
\[ = \frac{1}{\frac{4}{16} - \frac{1}{2} + 1} = \frac{16}{12} = \frac{4}{3}\]
APPEARS IN
RELATED QUESTIONS
f (x) = \[-\] | x + 1 | + 3 on R .
f(x) = (x \[-\] 5)4.
f(x) = x3 (x \[-\] 1)2 .
f(x) = (x \[-\] 1) (x+2)2.
f(x) = sin x \[-\] cos x, 0 < x < 2\[\pi\] .
f(x) =\[\frac{x}{2} + \frac{2}{x} , x > 0\] .
f(x) = x3\[-\] 6x2 + 9x + 15
`f(x) = (x+1) (x+2)^(1/3), x>=-2` .
`f(x)=xsqrt(32-x^2), -5<=x<=5` .
Show that \[\frac{\log x}{x}\] has a maximum value at x = e ?
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) = 2x3 − 15x2 + 36x + 1 on the interval [1, 5].
A square piece of tin of side 18 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Find this maximum volume.
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?
A large window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 metres find the dimensions of the rectangle will produce the largest area of the window.
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 point on the curve y2 = 4x which is nearest to the point (2,\[-\] 8).
Find the coordinates of a point on the parabola y=x2+7x + 2 which is closest to the strainght line y = 3x \[-\] 3 ?
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.
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 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 sufficient conditions for a point x = c to be a point of local maximum.
Write the maximum value of f(x) = x1/x.
If \[ax + \frac{b}{x} \frac{>}{} c\] for all positive x where a,b,>0, then _______________ .
Let f(x) = x3+3x2 \[-\] 9x+2. Then, f(x) has _________________ .
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 least value of the function f(x) = \[x3 - 18x2 + 96x\] in the interval [0,9] is _____________ .
If x+y=8, then the maximum value of xy is ____________ .
If(x) = x+\[\frac{1}{x}\],x > 0, then its greatest value 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:-
