Advertisements
Advertisements
प्रश्न
By completing the following activity, examine the function f(x) = x3 – 9x2 + 24x for maxima and minima
Solution: f(x) = x3 – 9x2 + 24x
∴ f'(x) = `square`
∴ f''(x) = `square`
For extreme values, f'(x) = 0, we get
x = `square` or `square`
∴ f''`(square)` = – 6 < 0
∴ f(x) is maximum at x = 2.
∴ Maximum value = `square`
∴ f''`(square)` = 6 > 0
∴ f(x) is maximum at x = 4.
∴ Minimum value = `square`
उत्तर
f(x) = x3 – 9x2 + 24x
∴ f'(x) = 3x2 – 18x + 24
∴ f''(x) = 6x – 18
For extreme values, f'(x) = 0, we get
3x2 – 18x + 24
∴ x2 – 6x + 8 = 0
∴ x2 – 4x – 2x + 8 = 0
∴ x(x – 4) – 2(x – 4) = 0
∴ (x – 4)(x – 2) = 0
x = 2 or 4
∴ f''(2) = 6(2) – 18
= 12 – 18
= – 6 < 0
∴ f(x) is maximum at x = 2.
∴ Maximum value = f(2)
= (2)3 – 9(2)2 + 24(2)
= 8 – 36 + 48
= 20
∴ f''(4) = 6(4) – 18
= 24 – 18
= 6 > 0
∴ f(x) is maximum at x = 4.
∴ Minimum value = f(4)
= (4)3 – 9(4)2 + 24(4)
= 64 – 144 + 96
= 16
APPEARS IN
संबंधित प्रश्न
Find the maximum and minimum value, if any, of the following function given by f(x) = 9x2 + 12x + 2
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:
g(x) = x3 − 3x
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 the maximum value of 2x3 − 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [−3, −1].
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is \[\pi : (\pi + 2)\].
Find the maximum and minimum of the following functions : f(x) = x log x
Divide the number 20 into two parts such that sum of their squares is minimum.
Show that among rectangles of given area, the square has least perimeter.
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.
The two parts of 120 for which the sum of double of first and square of second part is minimum, are ______.
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.
Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.
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.
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 ______.
The minimum value of 2sinx + 2cosx is ______.
A rectangle with one side lying along the x-axis is to be inscribed in the closed region of the xy plane bounded by the lines y = 0, y = 3x and y = 30 – 2x. The largest area of such a rectangle is ______.
Find the maximum profit that a company can make, if the profit function is given by P(x) = 72 + 42x – x2, where x is the number of units and P is the profit in rupees.
A box with a square base is to have an open top. The surface area of box is 147 sq. cm. What should be its dimensions in order that the volume is largest?
