Advertisements
Advertisements
Question
Find the maximum and minimum of the following functions : f(x) = x log x
Advertisements
Solution
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
RELATED QUESTIONS
If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is π/3.
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`. Also find maximum volume in terms of volume of the sphere
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 function given by f(x) = |x + 2| − 1.
Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 − 72x − 18x2.
At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?
What is the maximum value of the function sin x + cos x?
Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.
A square piece of tin of side 18 cm is to made into a box without a top by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?
Show that the right circular cone of least curved surface and given volume has an altitude equal to `sqrt2` time the radius of the base.
For all real values of x, the minimum value of `(1 - x + x^2)/(1+x+x^2)` is ______.
A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle.
Show that the minimum length of the hypotenuse is `(a^(2/3) + b^(2/3))^(3/2).`
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
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?
Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is \[\cot^{- 1} \left( \sqrt{2} \right)\] .
Find the maximum and minimum of the following functions : f(x) = `logx/x`
An open cylindrical tank whose base is a circle is to be constructed of metal sheet so as to contain a volume of `pia^3`cu cm of water. Find the dimensions so that the quantity of the metal sheet required is minimum.
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?
Solve the following:
A rectangular sheet of paper of fixed perimeter with the sides having their lengths in the ratio 8 : 15 converted into an open rectangular box by folding after removing the squares of equal area from all corners. If the total area of the removed squares is 100, the resulting box has maximum volume. Find the lengths of the rectangular sheet of paper.
Determine the maximum and minimum value of the following function.
f(x) = x log x
The total cost of producing x units is ₹ (x2 + 60x + 50) and the price is ₹ (180 − x) per unit. For what units is the profit maximum?
State whether the following statement is True or False:
An absolute maximum must occur at a critical point or at an end point.
A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.
Max value of z equals 3x + 2y subject to x + y ≤ 3, x ≤ 2, -2x + y ≤ 1, 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.
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.
Find the points of local maxima, local minima and the points of inflection of the function f(x) = x5 – 5x4 + 5x3 – 1. Also find the corresponding local maximum and local minimum values.
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.
The curves y = 4x2 + 2x – 8 and y = x3 – x + 13 touch each other at the point ______.
Find both the maximum and minimum values respectively of 3x4 - 8x3 + 12x2 - 48x + 1 on the interval [1, 4].
The function f(x) = x5 - 5x4 + 5x3 - 1 has ____________.
Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.
A ball is thrown upward at a speed of 28 meter per second. What is the speed of ball one second before reaching maximum height? (Given that g= 10 meter per second2)
A function f(x) is maximum at x = a when f'(a) > 0.
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 ______.
The minimum value of α for which the equation `4/sinx + 1/(1 - sinx)` = α has at least one solution in `(0, π/2)` is ______.
If y = alog|x| + bx2 + x has its extremum values at x = –1 and x = 2, then ______.
Let f(x) = (x – a)ng(x) , where g(n)(a) ≠ 0; n = 0, 1, 2, 3.... then ______.
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 ______.
The lateral edge of a regular rectangular pyramid is 'a' cm long. The lateral edge makes an angle a. with the plane of the base. The value of a for which the volume of the pyramid is greatest, is ______.
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.
Complete the following activity to divide 84 into two parts such that the product of one part and square of the other is maximum.
Solution: Let one part be x. Then the other part is 84 - x
Letf (x) = x2 (84 - x) = 84x2 - x3
∴ f'(x) = `square`
and f''(x) = `square`
For extreme values, f'(x) = 0
∴ x = `square "or" square`
f(x) attains maximum at x = `square`
Hence, the two parts of 84 are 56 and 28.
The shortest distance between the line y - x = 1and the curve x = y2 is
If \[f\] has a maximum value at \[c\], what is \[c\] called?
When is a point in the domain of a function called a critical point?
Which statement gives the meaning of a local minimum at \[c\]?
Assume \[f'(c)=0\] and the second derivative exists at \[c\]. Which condition gives a local minimum?
For \[f(x)=3x^4+4x^3-12x^2+12\], what is \[f''(x)\]?
