Advertisements
Advertisements
Question
The maximum value of f(x) = \[\frac{x}{4 - x + x^2}\] on [ \[-\] 1, 1] is _______________ .
Options
\[ \frac{1}{4}\]
\[- \frac{1}{3}\]
\[\frac{1}{6}\]
\[\frac{1}{5}\]
Advertisements
Solution
\[\text { Given: } f\left( x \right) = \frac{x}{4 - x + x^2}\]
\[ \Rightarrow f'\left( x \right) = \frac{4 - x + x^2 - x\left( - 1 + 2x \right)}{\left( 4 - x + x^2 \right)^2}\]
\[\text { For a local maxima or a local minima, we must have } \]
\[f'\left( x \right) = 0\]
\[ \Rightarrow \frac{4 - x + x^2 - x\left( - 1 + 2x \right)}{\left( 4 - x + x^2 \right)^2} = 0\]
\[ \Rightarrow 4 - x + x^2 - x\left( - 1 + 2x \right) = 0\]
\[ \Rightarrow 4 - x + x^2 + x - 2 x^2 = 0\]
\[ \Rightarrow x^2 = 4\]
\[ \Rightarrow x =\pm 2 \not\in \left( - 1, 1 \right)\]
\[\text { So,} \]
\[f\left( - 1 \right) = \frac{- 1}{4 - \left( - 1 \right) + \left( - 1 \right)^2} = \frac{- 1}{6}\]
\[f\left( 1 \right) = \frac{1}{4 - 1 + 1^2} = \frac{1}{4}\]
\[\text { Hence, the maximum value is } \frac{1}{4} . \]
APPEARS IN
RELATED QUESTIONS
f(x) = - (x-1)2+2 on R ?
f (x) = \[-\] | x + 1 | + 3 on R .
f(x) = x3 (x \[-\] 1)2 .
f(x) = (x \[-\] 1) (x+2)2.
f(x) = \[\frac{1}{x^2 + 2}\] .
f(x) = sin 2x, 0 < x < \[\pi\] .
f(x) = cos x, 0 < x < \[\pi\] .
`f(x)=sin2x-x, -pi/2<=x<=pi/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) = (x - 1) (x + 2)2.
f(x) = (x \[-\] 1) (x \[-\] 2)2.
Find the maximum and minimum values of the function f(x) = \[\frac{4}{x + 2} + x .\]
Find the maximum and minimum values of y = tan \[x - 2x\] .
f(x) = 4x \[-\] \[\frac{x^2}{2}\] in [ \[-\] 2,4,5] .
Divide 64 into two parts such that the sum of the cubes of two parts is minimum.
Divide 15 into two parts such that the square of one multiplied with the cube of the other 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.
Show that the height of the cylinder of maximum volume that can be inscribed a sphere of radius R is \[\frac{2R}{\sqrt{3}} .\]
Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm ?
A closed cylinder has volume 2156 cm3. What will be the radius of its base so that its total surface area is minimum ?
Find the maximum slope of the curve y = \[- x^3 + 3 x^2 + 2x - 27 .\]
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 ?
Write necessary condition for a point x = c to be an extreme point of the function f(x).
Write the minimum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
Write the minimum value of f(x) = xx .
If \[ax + \frac{b}{x} \frac{>}{} c\] for all positive x where a,b,>0, then _______________ .
The minimum value of \[\frac{x}{\log_e x}\] is _____________ .
For the function f(x) = \[x + \frac{1}{x}\]
The number which exceeds its square by the greatest possible quantity is _________________ .
The sum of two non-zero numbers is 8, the minimum value of the sum of the reciprocals is ______________ .
If x lies in the interval [0,1], then the least value of x2 + x + 1 is _______________ .
The least and greatest values of f(x) = x3\[-\] 6x2+9x in [0,6], are ___________ .
The minimum value of \[\left( x^2 + \frac{250}{x} \right)\] is __________ .
Let x, y be two variables and x>0, xy=1, then minimum value of x+y is _______________ .
f(x) = 1+2 sin x+3 cos2x, `0<=x<=(2pi)/3` is ________________ .
The function f(x) = \[2 x^3 - 15 x^2 + 36x + 4\] is maximum at x = ________________ .
The maximum value of f(x) = \[\frac{x}{4 + x + x^2}\] on [ \[-\] 1,1] is ___________________ .
A wire of length 34 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a rectangle whose length is twice its breadth. What should be the lengths of the two pieces, so that the combined area of the square and the rectangle is minimum?
