Advertisements
Advertisements
प्रश्न
Find the maximum and minimum of the following functions : f(x) = x log x
Advertisements
उत्तर
f(x) = x log x
∴ f'(x) = `d/dx (x log x)`
= `x.d/dx (log x) + logx.d/dx(x)`
= `x xx (1)/x + (log x) xx 1`
= 1 + log x
and
f"(x) = `d/dx(1 + log x)`
= `0 + (1)/x = (1)/x`
Now, f'(x) = 0, if 1 + log x = 0
i.e. if log x = – 1 = – log e
i.e. if log x = `log(e^-1) = log(1)/e`
i.e. if x = `(1)/e`
When `x = (1)e, f"(x) = (1)/((1/e)` = e > 0
∴ by the second derivative test, f is minimum at x = `(1)/e`.
Minimum value of f at x = `(1)/e`
= `(1)/elog(1/e)`
= `(1)/e.log(e^-1)`
= `(1)/e.(-1)log e`
= `-(1)/e`. ...[∵ log e = 1]
APPEARS IN
संबंधित प्रश्न
Examine the maxima and minima of the function f(x) = 2x3 - 21x2 + 36x - 20 . Also, find the maximum and minimum values of f(x).
Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]
Find the maximum and minimum value, if any, of the following function given by f(x) = −(x − 1)2 + 10
Find the maximum and minimum value, if any, of the following function given by h(x) = sin(2x) + 5.
Find the maximum and minimum value, if any, of the following function given by f(x) = |sin 4x + 3|
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
f(x) = x2
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
`g(x) = x/2 + 2/x, x > 0`
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
`f(x) = 4x - 1/x x^2, x in [-2 ,9/2]`
Find both the maximum value and the minimum value of 3x4 − 8x3 + 12x2 − 48x + 25 on the interval [0, 3].
For all real values of x, the minimum value of `(1 - x + x^2)/(1+x+x^2)` is ______.
A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening
Find the points at which the function f given by f (x) = (x – 2)4 (x + 1)3 has
- local maxima
- local minima
- point of inflexion
The volume of a closed rectangular metal box with a square base is 4096 cm3. The cost of polishing the outer surface of the box is Rs. 4 per cm2. Find the dimensions of the box for the minimum cost of polishing it.
Find the maximum and minimum of the following functions : f(x) = `logx/x`
Divide the number 20 into two parts such that sum of their squares is minimum.
Show that among rectangles of given area, the square has least perimeter.
Solve the following : Show that the height of a right circular cylinder of greatest volume that can be inscribed in a right circular cone is one-third of that of the cone.
Solve the following : Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/(3)`.
Determine the maximum and minimum value of the following function.
f(x) = x log x
A wire of length 120 cm is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum
AB is a diameter of a circle and C is any point on the circle. Show that the area of ∆ABC is maximum, when it is isosceles.
If x is real, the minimum value of x2 – 8x + 17 is ______.
The smallest value of the polynomial x3 – 18x2 + 96x in [0, 9] is ______.
The maximum value of sin x . cos x is ______.
The curves y = 4x2 + 2x – 8 and y = x3 – x + 13 touch each other at the point ______.
The maximum value of `["x"("x" − 1) + 1]^(1/3)`, 0 ≤ x ≤ 1 is:
Find all the points of local maxima and local minima of the function f(x) = (x - 1)3 (x + 1)2
If y = x3 + x2 + x + 1, then y ____________.
Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.
The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is ____________.
The distance of that point on y = x4 + 3x2 + 2x which is nearest to the line y = 2x - 1 is ____________.
The combined resistance R of two resistors R1 and R2 (R1, R2 > 0) is given by `1/"R" = 1/"R"_1 + 1/"R"_2`. If R1 + R2 = C (a constant), then maximum resistance R is obtained if ____________.
Let f(x) = 1 + 2x2 + 22x4 + …… + 210x20. Then f (x) has ____________.
The maximum value of `[x(x - 1) + 1]^(2/3), 0 ≤ x ≤ 1` is
The function g(x) = `(f(x))/x`, x ≠ 0 has an extreme value when ______.
Let x and y be real numbers satisfying the equation x2 – 4x + y2 + 3 = 0. If the maximum and minimum values of x2 + y2 are a and b respectively. Then the numerical value of a – b is ______.
Let f(x) = |(x – 1)(x2 – 2x – 3)| + x – 3, x ∈ R. If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M is equal to ______.
The sum of all the local minimum values of the twice differentiable function f : R `rightarrow` R defined by
f(x) = `x^3 - 3x^2 - (3f^('')(2))/2 x + f^('')(1)`
The minimum value of the function f(x) = xlogx is ______.
A rod AB of length 16 cm. rests between the wall AD and a smooth peg, 1 cm from the wall and makes an angle θ with the horizontal. The value of θ for which the height of G, the midpoint of the rod above the peg is minimum, is ______.
A metal wire of 36 cm long is bent to form a rectangle. Find its dimensions when its area is maximum.
Find the point on the curve y2 = 4x, which is nearest to the point (2, 1).
A box with a square base is to have an open top. The surface area of box is 147 sq. cm. What should be its dimensions in order that the volume is largest?
Which statement gives the meaning of a local minimum at \[c\]?
What is an absolute minimum?
For a continuous function on a closed interval \[[a,b]\], which values must be compared to find absolute maxima and minima?
For \[f(x)=3x^4+4x^3-12x^2+12\], which is the correct factorization of \[f'(x)\]?
