Advertisements
Advertisements
Question
Determine the maximum and minimum value of the following function.
f(x) = x log x
Advertisements
Solution
f(x) = x log x
∴ f'(x) =`"x" "d"/"dx" (log "x") + log "x" "d"/"dx" ("x")`
`= "x" xx 1/"x" + log "x" xx 1 = 1 + log "x"`
and f''(x) = `0 + 1/"x" = 1/"x"`
Consider, f'(x) = 0
∴ 1 + log x = 0
∴ log x = - 1
∴ log x = - log e = log e-1 = log `(1/"e")`
∴ x = `1/"e"`
For x = `1/"e"`
`f''(1/"e") = 1/(1/"e") = "e" > 0`
∴ f(x) attains minimum value at x = `1/"e"`.
∴ Minimum value = `"f"(1/"e") = 1/"e" log (1/"e") = 1/"e" log "e"^-1`
`= ((- 1)/"e") (1) = ((- 1)/"e")`
∴ The function f(x) has minimum value `(-1)/"e"` at x = `1/"e"`.
APPEARS IN
RELATED QUESTIONS
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 maximum and minimum value, if any, of the following function given by g(x) = − |x + 1| + 3.
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
f (x) = sin x + cos x , x ∈ [0, π]
Find both the maximum value and the minimum value of 3x4 − 8x3 + 12x2 − 48x + 25 on the interval [0, 3].
Find the maximum value of 2x3 − 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [−3, −1].
Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
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.
Find the maximum and minimum of the following functions : f(x) = `x^2 + (16)/x^2`
A ball is thrown in the air. Its height at any time t is given by h = 3 + 14t – 5t2. Find the maximum height it can reach.
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:
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.
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?
Max value of z equals 3x + 2y subject to x + y ≤ 3, x ≤ 2, -2x + y ≤ 1, x ≥ 0, y ≥ 0 is ______
Find all the points of local maxima and local minima of the function f(x) = `- 3/4 x^4 - 8x^3 - 45/2 x^2 + 105`
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 maximum value of sin x . cos x is ______.
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.
The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is ____________.
Find the area of the largest isosceles triangle having a perimeter of 18 meters.
Range of projectile will be maximum when angle of projectile is
The point on the curve `x^2 = 2y` which is nearest to the point (0, 5) is
The maximum value of `[x(x - 1) + 1]^(2/3), 0 ≤ x ≤ 1` is
Read the following passage and answer the questions given below.
|
In an elliptical sport field the authority wants to design a rectangular soccer field with the maximum possible area. The sport field is given by the graph of `x^2/a^2 + y^2/b^2` = 1. |
- If the length and the breadth of the rectangular field be 2x and 2y respectively, then find the area function in terms of x.
- Find the critical point of the function.
- Use First derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
OR
Use Second Derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
Determine the minimum value of the function.
f(x) = 2x3 – 21x2 + 36x – 20
The shortest distance between the line y - x = 1and the curve x = y2 is
The absolute maximum value of the function f(x) = 2x3 − 3x2 − 36x + 9 defined on [−3, 3] is ______.
