Advertisements
Advertisements
प्रश्न
For the function f(x) = \[x + \frac{1}{x}\]
पर्याय
x = 1 is a point of maximum
x = \[-\] 1 is a point of minimum
maximum value > minimum value
maximum value < minimum value
Advertisements
उत्तर
\[\text { maximum value < minimum value}\]
\[\text { Given:} f\left( x \right) = x + \frac{1}{x}\]
\[ \Rightarrow f'\left( x \right) = 1 - \frac{1}{x^2}\]
\[\text { For a local maxima or a local minima, we must have} \]
\[f'\left( x \right) = 0\]
\[ \Rightarrow 1 - \frac{1}{x^2} = 0\]
\[ \Rightarrow x^2 - 1 = 0\]
\[ \Rightarrow x^2 = 1\]
\[ \Rightarrow x = \pm 1\]
\[\text { Now }, \]
\[f''\left( x \right) = \frac{2}{x^3}\]
\[ \Rightarrow f''\left( 1 \right) = \frac{2}{1} = 2 > 0\]
\[\text { So, x = 1 is a local minima.}\]
\[\text { Also }, \]
\[f''\left( - 1 \right) = - 2 < 0\]
\[\text {So, x = - 1 is a localmaxima }.\]
\[\text { The local minimum value is given by }\]
\[f\left( 1 \right) = 2\]
\[\text { The local maximum value is given by }\]
\[f\left( - 1 \right) = - 2\]
\[ \therefore \text { Maximum value < Minimum value }\]
APPEARS IN
संबंधित प्रश्न
f(x) = x3 \[-\] 6x2 + 9x + 15 .
f(x) = sin 2x, 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) = x4 \[-\] 62x2 + 120x + 9.
f(x) = \[x^3 - 2a x^2 + a^2 x, a > 0, x \in R\] .
f(x) = \[x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}\] .
The function y = a log x+bx2 + x has extreme values at x=1 and x=2. Find a and b ?
Find the maximum and minimum values of the function f(x) = \[\frac{4}{x + 2} + x .\]
If f(x) = x3 + ax2 + bx + c has a maximum at x = \[-\] 1 and minimum at x = 3. Determine a, b and c ?
Find the maximum value of 2x3\[-\] 24x + 107 in the interval [1,3]. Find the maximum value of the same function in [ \[-\] 3, \[-\] 1].
Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.
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 rectangle is inscribed in a semi-circle of radius r with one of its sides on diameter of semi-circle. Find the dimension of the rectangle so that its area is maximum. Find also the area ?
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is \[6\sqrt{3}\]r.
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) ?
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 ?
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.
If f(x) attains a local minimum at x = c, then write the values of `f' (c)` and `f'' (c)`.
Write the minimum 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 .
Write the minimum value of f(x) = xx .
Let f(x) = x3+3x2 \[-\] 9x+2. Then, f(x) has _________________ .
The minimum value of f(x) = \[x4 - x2 - 2x + 6\] is _____________ .
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 _____________ .
The maximum value of f(x) = \[\frac{x}{4 - x + x^2}\] on [ \[-\] 1, 1] is _______________ .
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 ______________ .
If(x) = x+\[\frac{1}{x}\],x > 0, then its greatest value is _______________ .
If(x) = \[\frac{1}{4x^2 + 2x + 1}\] then its maximum value is _________________ .
Let x, y be two variables and x>0, xy=1, then minimum value of x+y is _______________ .
The maximum value of f(x) = \[\frac{x}{4 + x + x^2}\] on [ \[-\] 1,1] is ___________________ .
The minimum value of x loge x is equal to ____________ .
