Advertisements
Advertisements
Question
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
`f(x) = 4x - 1/x x^2, x in [-2 ,9/2]`
Advertisements
Solution
Given function f(x) = 4x `- 1/2 x^2,` interval `[-2,9/2]`
∴ f'(x) = 4 - `1/2`. 2x = 4 - x
If f'(x) = 0, then 4 - x = 0 ⇒ x = 4
At x = -2, f(-2) = 4 (-2) - `1/2 (-2)^2`
`= - 8 - 1/2 xx 4`
= - 8 - 2
= - 10
At x = 4, `f(4) = 4(4) - (4)^2/2`
`= 16 - 16/2`
= 16 - 8
= 8
At x = `9/2`, `f (9/2) = 4 xx 9/2 - 1/2 xx 81/4`
`= 18 - 81/8`
`= (144 - 81)/8`
`= 63/8`
= 7.875
∴ Absolute maximum value of f (x) = 8 at x = 4
Absolute minimum value of f (x) = -10 at x = -2
APPEARS IN
RELATED QUESTIONS
If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f(x).
Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]
Show that the height of the cylinder of maximum volume, that can be inscribed in a sphere of radius R is `(2R)/sqrt3.` Also, find the maximum volume.
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.
Find the maximum and minimum value, if any, of the following function given by g(x) = − |x + 1| + 3.
Prove that the following function do not have maxima or minima:
h(x) = x3 + x2 + x + 1
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 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?
Find the maximum area of an isosceles triangle inscribed in the ellipse `x^2/ a^2 + y^2/b^2 = 1` with its vertex at one end of the major axis.
A rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get the maximum area. Also, find the maximum area.
The profit function P(x) of a firm, selling x items per day is given by P(x) = (150 – x)x – 1625 . Find the number of items the firm should manufacture to get maximum profit. Find the maximum profit.
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 : An open box with a square base is to be made out of given quantity of sheet of area a2. Show that the maximum volume of the box is `a^3/(6sqrt(3)`.
Solve the following : 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)`.
Determine the maximum and minimum value of the following function.
f(x) = 2x3 – 21x2 + 36x – 20
Divide the number 20 into two parts such that their product is maximum.
A rectangular sheet of paper has it area 24 sq. Meters. The margin at the top and the bottom are 75 cm each and the sides 50 cm each. What are the dimensions of the paper if the area of the printed space is maximum?
Show that the function f(x) = 4x3 – 18x2 + 27x – 7 has neither maxima nor minima.
A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs 300/- per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Re 1/- one subscriber will discontinue the service. Find what increase will bring maximum profit?
The maximum value of sin x . cos x is ______.
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
Find the maximum profit that a company can make, if the profit function is given by P(x) = 41 + 24x – 18x2.
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 f(x) = (x – a)ng(x) , where g(n)(a) ≠ 0; n = 0, 1, 2, 3.... then ______.
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 ______.
Let f(x) = |(x – 1)(x2 – 2x – 3)| + x – 3, x ∈ R. If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M is equal to ______.
The maximum value of z = 6x + 8y subject to constraints 2x + y ≤ 30, x + 2y ≤ 24 and x ≥ 0, y ≥ 0 is ______.
The maximum value of f(x) = `logx/x (x ≠ 0, x ≠ 1)` is ______.
Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.
If f(x) = `1/(4x^2 + 2x + 1); x ∈ R`, then find the maximum value of f(x).
If Mr. Rane order x chairs at the price p = (2x2 - 12x - 192) per chair. How many chairs should he order so that the cost of deal is minimum?
Solution: Let Mr. Rane order x chairs.
Then the total price of x chairs = p·x = (2x2 - 12x- 192)x
= 2x3 - 12x2 - 192x
Let f(x) = 2x3 - 12x2 - 192x
∴ f'(x) = `square` and f''(x) = `square`
f'(x ) = 0 gives x = `square` and f''(8) = `square` > 0
∴ f is minimum when x = 8
Hence, Mr. Rane should order 8 chairs for minimum cost of deal.
For a function defined on an interval \[I\], which condition means that \[f\] has a maximum value at \[c\in I\]?
If \[f\] has a maximum value at \[c\], what is \[c\] called?
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 \[c\]?
Why must the left endpoint \[a\] and right endpoint \[b\] be checked for absolute extrema on \[[a,b]\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], what is \[f''(x)\]?
