Advertisements
Advertisements
प्रश्न
The maximum value of `(1/x)^x` is ______.
पर्याय
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
Examine the maxima and minima of the function f(x) = 2x3 - 21x2 + 36x - 20 . Also, find the maximum and minimum values of f(x).
If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f(x).
Find the maximum and minimum value, if any, of the following function given by h(x) = x + 1, x ∈ (−1, 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 the absolute maximum value and the absolute minimum value of the following function in the given interval:
f (x) = (x −1)2 + 3, x ∈[−3, 1]
Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 − 72x − 18x2.
Find both the maximum value and the minimum value of 3x4 − 8x3 + 12x2 − 48x + 25 on the interval [0, 3].
What is the maximum value of the function sin x + cos x?
Find the maximum and minimum values of x + sin 2x on [0, 2π].
Find two numbers whose sum is 24 and whose product is as large as possible.
A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `tan^(-1) sqrt(2)`
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when the depth of the tank is half of its width. If the cost is to be borne by nearby settled lower-income families, for whom water will be provided, what kind of value is hidden in this question?
Show that a cylinder of a given volume, which is open at the top, has minimum total surface area when its height is equal to the radius of its base.
A rod of 108 meters long is bent to form a rectangle. Find its dimensions if the area is maximum. Let x be the length and y be the breadth of the rectangle.
Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.
Find the maximum and minimum of the following functions : f(x) = x3 – 9x2 + 24x
A box with a square base is to have an open top. The surface area of the box is 192 sq cm. What should be its dimensions in order that the volume is largest?
Show that the height of a closed right circular cylinder of given volume and least surface area is equal to its diameter.
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.
Examine the function for maxima and minima f(x) = x3 - 9x2 + 24x
Find the local maximum and local minimum value of f(x) = x3 − 3x2 − 24x + 5
If f(x) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.
If R is the circum radius of Δ ABC, then A(Δ ABC) = ______.
The function y = 1 + sin x is maximum, when x = ______
The maximum and minimum values for the function f(x) = 4x3 - 6x2 on [-1, 2] are ______
An open box with square base is to be made of a given quantity of cardboard of area c2. Show that the maximum volume of the box is `"c"^3/(6sqrt(3))` cubic units
A metal box with a square base and vertical sides is to contain 1024 cm3. The material for the top and bottom costs Rs 5/cm2 and the material for the sides costs Rs 2.50/cm2. Find the least cost of the box.
Find the local minimum value of the function f(x) `= "sin"^4" x + cos"^4 "x", 0 < "x" < pi/2`
The maximum value of `[x(x - 1) + 1]^(2/3), 0 ≤ x ≤ 1` is
A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is ______.
Let A = [aij] be a 3 × 3 matrix, where
aij = `{{:(1, "," if "i" = "j"),(-x, "," if |"i" - "j"| = 1),(2x + 1, "," "otherwise"):}`
Let a function f: R→R be defined as f(x) = det(A). Then the sum of maximum and minimum values of f on R is equal to ______.
If S1 and S2 are respectively the sets of local minimum and local maximum points of the function. f(x) = 9x4 + 12x3 – 36x2 + 25, x ∈ R, then ______.
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 ______.
Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.
Divide the number 100 into two parts so that the sum of their squares is minimum.
