Advertisements
Advertisements
Question
Divide 20 into two ports, so that their product is maximum.
Advertisements
Solution
Let one part of 20 be x.
∴ The other part is (20 – x)
∴ Product = x.(20 – x)
Which has to be maximized.
∴ f(x) = x.(20 – x)
= 20x – x2
∴ f'(x) = 20 – 2x
f"(x) = – 2 < 0
Let f'(x) = 0
∴ 20 – 2x = 0
⇒ 2x = 20
⇒ x = 10
And f"(x) = – 2 < 0
∴ By 2nd derivative test, f is maximum at x = 10
∴ 20 – x = 20 – 10 = 10.
∴ The required parts of 20 are 10 and 10.
APPEARS IN
RELATED QUESTIONS
An open box is to be made out of a piece of a square card board of sides 18 cms by cutting off equal squares from the comers and turning up the sides. Find the maximum volume of the box.
Find the maximum and minimum value, if any, of the following function given by g(x) = − |x + 1| + 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) = sinx − cos x, 0 < x < 2π
Show that semi-vertical angle of right circular cone of given surface area and maximum volume is `Sin^(-1) (1/3).`
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
Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
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.
Find the maximum and minimum of the following functions : f(x) = `x^2 + (16)/x^2`
Divide the number 20 into two parts such that sum of their squares is minimum.
Find the largest size of a rectangle that can be inscribed in a semicircle of radius 1 unit, so that two vertices lie on the diameter.
Determine the maximum and minimum value of the following function.
f(x) = `x^2 + 16/x`
The minimum value of Z = 5x + 8y subject to x + y ≥ 5, 0 ≤ x ≤ 4, y ≥ 2, x ≥ 0, y ≥ 0 is ____________.
The minimum value of the function f(x) = 13 - 14x + 9x2 is ______
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 the local minimum value of the function f(x) `= "sin"^4" x + cos"^4 "x", 0 < "x" < pi/2`
Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.
A function f(x) is maximum at x = a when f'(a) > 0.
A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is k (meter), then `(4/π + 1)`k is equal to ______.
The function g(x) = `(f(x))/x`, x ≠ 0 has an extreme value when ______.
The set of values of p for which the points of extremum of the function f(x) = x3 – 3px2 + 3(p2 – 1)x + 1 lie in the interval (–2, 4), is ______.
The greatest value of the function f(x) = `tan^-1x - 1/2logx` in `[1/sqrt(3), sqrt(3)]` is ______.
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 2sinx + 2cosx is ______.
Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.
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"`
A maximum or minimum value of \[f\] is called a(n) _____.
Let \[c\] be a critical point of a continuous function \[f\]. If \[f'(x)\] changes sign from positive to negative as \[x\] passes through \[c\], what is \[c\]?
