Advertisements
Advertisements
प्रश्न
The maximum value of `(1/x)^x` is ______.
विकल्प
e
ex
`"e"^(1/"e")`
`(1/"e")^(1/"e")`
ee
Advertisements
उत्तर
The maximum value of `(1/x)^x` is `underlinebb(e^(1/e))`.
Explanation:
Let f(x) = `(1/x)^x`
Taking log on both sides, we get
log [f (x)] = `x log 1/x`
⇒ log [f (x)] = `x log x^-1`
⇒ log [f (x)] = – [x log x]
Differentiating both sides w.r.t. x, we get
`1/("f"(x)) * "f'"(x) = - [x * 1/x + log x * 1]`
= `- "f"(x) [1 + log x]`
⇒ f'(x) = `- (1/x)^x [1 + log x]`
For local maxima and local minima f'(x) = 0
`-(1/x)^x [1 + log x]` = 0
⇒ `(1/x)^x [1 + log x]`= 0
`(1/x)^x ≠ 0`
∴ 1 + log x = 0
⇒ log x = – 1
⇒ x = e–1
So, x = `1/"e"` is the stationary point.
Now f'(x) = `-(1/x)^x [1 + log x]`
f"(x) = `-[(1/x)^x (1/x) + (1 + log x) * "d"/"dx" (x)^x]`
f"(x) = `-[("e")^(1/"e") ("e") + (1 + log 1/"e") "d"/"dx" (1/"e")^(1/"e")]`
x = `1/"e"`
= `-"e"^(1/"e") 1 < 0` maxima
∴ Maximum value of the function at x = `1/"e"` is
`"f"(1/"e") = (1/(1/"e"))^(1/"e") = "e"^(1/"e")`
संबंधित प्रश्न
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 g(x) = x3 + 1.
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 two numbers whose sum is 24 and whose product is as large as possible.
Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?
Show that semi-vertical angle of right circular cone of given surface area and maximum volume is `Sin^(-1) (1/3).`
Show that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to \[ \frac{2}{3} \] of the diameter of the sphere.
A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is \[\pi : (\pi + 2)\].
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) = `x^2 + (16)/x^2`
Find the maximum and minimum of the following functions : f(x) = x log x
Solve the following:
Find the maximum and minimum values of the function f(x) = cos2x + sinx.
Find the local maximum and local minimum value of f(x) = x3 − 3x2 − 24x + 5
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
A rectangular sheet of paper has it area 24 sq. Meters. The margin at the top and the bottom are 75 cm each and the sides 50 cm each. What are the dimensions of the paper if the area of the printed space is maximum?
The minimum value of Z = 5x + 8y subject to x + y ≥ 5, 0 ≤ x ≤ 4, y ≥ 2, x ≥ 0, y ≥ 0 is ____________.
If f(x) = `x + 1/x, x ne 0`, then local maximum and x minimum values of function f are respectively.
The maximum value of function x3 - 15x2 + 72x + 19 in the interval [1, 10] is ______.
The maximum and minimum values for the function f(x) = 4x3 - 6x2 on [-1, 2] are ______
If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between them is `pi/3`
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.
The sum of the surface areas of a rectangular parallelopiped with sides x, 2x and `x/3` and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of the sphere. Also find the minimum value of the sum of their volumes.
The maximum value of `["x"("x" − 1) + 1]^(1/3)`, 0 ≤ x ≤ 1 is:
Find the local minimum value of the function f(x) `= "sin"^4" x + cos"^4 "x", 0 < "x" < pi/2`
The coordinates of the point on the parabola y2 = 8x which is at minimum distance from the circle x2 + (y + 6)2 = 1 are ____________.
Read the following passage and answer the questions given below.
|
|
- Is the function differentiable in the interval (0, 12)? Justify your answer.
- If 6 is the critical point of the function, then find the value of the constant m.
- Find the intervals in which the function is strictly increasing/strictly decreasing.
OR
Find the points of local maximum/local minimum, if any, in the interval (0, 12) as well as the points of absolute maximum/absolute minimum in the interval [0, 12]. Also, find the corresponding local maximum/local minimum and the absolute ‘maximum/absolute minimum values of the function.
Let f: R → R be a function defined by f(x) = (x – 3)n1(x – 5)n2, n1, n2 ∈ N. Then, which of the following is NOT true?
The range of a ∈ R for which the function f(x) = `(4a - 3)(x + log_e5) + 2(a - 7)cot(x/2)sin^2(x/2), x ≠ 2nπ, n∈N` has critical points, is ______.
If the function y = `(ax + b)/((x - 4)(x - 1))` has an extremum at P(2, –1), then the values of a and b are ______.
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)`
Check whether the function f : R `rightarrow` R defined by f(x) = x3 + x, has any critical point/s or not ? If yes, then find the point/s.
The rectangle has area of 50 cm2. Complete the following activity to find its dimensions for least perimeter.
Solution: Let x cm and y cm be the length and breadth of a rectangle.
Then its area is xy = 50
∴ `y =50/x`
Perimeter of rectangle `=2(x+y)=2(x+50/x)`
Let f(x) `=2(x+50/x)`
Then f'(x) = `square` and f''(x) = `square`
Now,f'(x) = 0, if x = `square`
But x is not negative.
∴ `x = root(5)(2) "and" f^('')(root(5)(2))=square>0`
∴ by the second derivative test f is minimum at x = `root(5)(2)`
When x = `root(5)(2),y=50/root(5)(2)=root(5)(2)`
∴ `x=root(5)(2) "cm" , y = root(5)(2) "cm"`
Hence, rectangle is a square of side `root(5)(2) "cm"`
Find the point on the curve y2 = 4x, which is nearest to the point (2, 1).
