Advertisements
Advertisements
प्रश्न
Find the maximum and minimum of the following functions : f(x) = x3 – 9x2 + 24x
Advertisements
उत्तर
f(x) = x3 – 9x2 + 24x
∴ f'(x) = `d/dx(x^3 - 9x^2 + 24x)`
= 3x2 – 9 x 2x + 24 x 1
= 3x2 – 18x + 24
and
f''(x) = `d/dx(3x^2 - 18x + 24)`
= 3 x 2x – 18 x 1 + 0
= 6x – 18
f''(x) = 0 gives 3x2 – 18x + 24 = 0
∴ x2 – 6 x + 8 = 0
∴ (x – 2)(x – 4) = 0
∴ the roots of f'(x) = 0 are x1 = 2 and x2 = 4.
(a) f"(2)
= 6(2) – 18
= – 6 < 0
∴ by the second derivaative test, f has maximum at x = 2 and maximum value of at x = 2
= f(2)
= (2)3 – 9(2)2 + 24(2)
= 8 – 36 + 48
= 20
(b) f"(4) = 6(4) – 18 = 6 > 0
∴ by the second derivaative test, f has maximum at x = 4 and maximum value of at x = 4
= f(4)
= (4)3 – 9(4)2 + 24(4)
= 64 – 144 + 96
= 16.
Hence, the function f has maximum value 20 at x = 2 and minimum value 16 at x = 4.
APPEARS IN
संबंधित प्रश्न
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 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:
`h(x) = sinx + cosx, 0 < x < pi/2`
Find both the maximum value and the minimum value of 3x4 − 8x3 + 12x2 − 48x + 25 on the interval [0, 3].
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.
Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.
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?
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
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
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 absolute maximum and minimum values of the function f given by f (x) = cos2 x + sin x, x ∈ [0, π].
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 per cm2 and the material for the sides costs Rs 2.50 per cm2. Find the least cost of the box
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 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?
Choose the correct option from the given alternatives :
If f(x) = `(x^2 - 1)/(x^2 + 1)`, for every real x, then the minimum value of f is ______.
Solve the following : Show that a closed right circular cylinder of given surface area has maximum volume if its height equals the diameter of its base.
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) = 2x3 – 21x2 + 36x – 20
Determine the maximum and minimum value of the following function.
f(x) = x log x
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 metal wire of 36 cm long is bent to form a rectangle. By completing the following activity, find it’s dimensions when it’s area is maximum.
Solution: Let the dimensions of the rectangle be x cm and y cm.
∴ 2x + 2y = 36
Let f(x) be the area of rectangle in terms of x, then
f(x) = `square`
∴ f'(x) = `square`
∴ f''(x) = `square`
For extreme value, f'(x) = 0, we get
x = `square`
∴ f''`(square)` = – 2 < 0
∴ Area is maximum when x = `square`, y = `square`
∴ Dimensions of rectangle are `square`
The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is ______.
The minimum value of Z = 5x + 8y subject to x + y ≥ 5, 0 ≤ x ≤ 4, y ≥ 2, x ≥ 0, y ≥ 0 is ____________.
Max value of z equals 3x + 2y subject to x + y ≤ 3, x ≤ 2, -2x + y ≤ 1, x ≥ 0, y ≥ 0 is ______
Twenty meters of wire is available for fencing off a flowerbed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is ______
Show that the function f(x) = 4x3 – 18x2 + 27x – 7 has neither maxima nor minima.
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.
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.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.
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.
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 ______.
If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to ______.
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 ______.
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 ______.
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 volume of the greatest cylinder which can be inscribed in a cone of height 30 cm and semi-vertical angle 30° is ______.
A straight line is drawn through the point P(3, 4) meeting the positive direction of coordinate axes at the points A and B. If O is the origin, then minimum area of ΔOAB is equal to ______.
If f(x) = `1/(4x^2 + 2x + 1); x ∈ R`, then find the maximum value of f(x).
A metal wire of 36 cm long is bent to form a rectangle. Find its dimensions when its area is maximum.
Which statement gives the meaning of a local minimum at \[c\]?
What is an absolute minimum?
Let \[c\] be a critical point of a continuous function \[f\]. If \[f'(x)\] changes sign from negative to positive as \[x\] passes through \[c\], what is \[c\]?
What are the critical points obtained from \[12x(x-1)(x+2)=0\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], what conclusion follows at \[x=0\]?
