Advertisements
Advertisements
Question
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`
Advertisements
Solution
f′(x) = –3x3 – 24x2 – 45x
= – 3x(x2 + 8x + 15)
= – 3x(x + 5)(x + 3)
f′(x) = 0
⇒ x = –5, x = –3, x = 0
f″(x) = –9x2 – 48x – 45
= –3(3x2 + 16x + 15)
f″(0) = – 45 < 0. Therefore, x = 0 is point of local maxima
f″(–3) = 18 > 0. Therefore, x = –3 is point of local minima
f″(–5) = –30 < 0. Therefore x = –5 is point of local maxima.
APPEARS IN
RELATED QUESTIONS
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.
A telephone company in a town has 5000 subscribers on its list and collects fixed rent charges of Rs.3,000 per year from each subscriber. The company proposes to increase annual rent and it is believed that for every increase of one rupee in the rent, one subscriber will be discontinued. Find what increased annual rent will bring the maximum annual income to the company.
Find the maximum and minimum value, if any, of the following function given by g(x) = x3 + 1.
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
f (x) = (x −1)2 + 3, x ∈[−3, 1]
Find both the maximum value and the minimum value of 3x4 − 8x3 + 12x2 − 48x + 25 on the interval [0, 3].
Find two numbers whose sum is 24 and whose product is as large as possible.
Find two positive numbers x and y such that their sum is 35 and the product x2y5 is a maximum.
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
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
A rod of 108 meters long is bent to form a rectangle. Find its dimensions if the area is maximum. Let x be the length and y be the breadth of the rectangle.
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?
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 : A window is in the form of a rectangle surmounted by a semicircle. If the perimeter be 30 m, find the dimensions so that the greatest possible amount of light may be admitted.
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.
State whether the following statement is True or False:
An absolute maximum must occur at a critical point or at an end point.
The function f(x) = x log x is minimum at 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
If f(x) = px5 + qx4 + 5x3 - 10 has local maximum and minimum at x = 1 and x = 3 respectively then (p, q) = ______.
Max value of z equals 3x + 2y subject to x + y ≤ 3, x ≤ 2, -2x + y ≤ 1, x ≥ 0, y ≥ 0 is ______
If f(x) = `x + 1/x, x ne 0`, then local maximum and x minimum values of function f are respectively.
The maximum value of `(1/x)^x` is ______.
Find the local minimum value of the function f(x) `= "sin"^4" x + cos"^4 "x", 0 < "x" < pi/2`
If y `= "ax - b"/(("x" - 1)("x" - 4))` has a turning point P(2, -1), then find the value of a and b respectively.
The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is ____________.
The function `"f"("x") = "x" + 4/"x"` has ____________.
Let f(x) = 1 + 2x2 + 22x4 + …… + 210x20. Then f (x) has ____________.
A ball is thrown upward at a speed of 28 meter per second. What is the speed of ball one second before reaching maximum height? (Given that g= 10 meter per second2)
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.
Let f: R → R be a function defined by f(x) = (x – 3)n1(x – 5)n2, n1, n2 ∈ N. Then, which of the following is NOT true?
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 ______.
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 minimum value of 2sinx + 2cosx is ______.
Sum of two numbers is 5. If the sum of the cubes of these numbers is least, then find the sum of the squares of these numbers.
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.
If \[\mathrm{A}+\mathrm{B}=\frac{\pi}{2}\] then the maximum value of cosA.cosB is
If \[f\] has a maximum value at \[c\], what is \[c\] called?
A maximum or minimum value of \[f\] is called a(n) _____.
Which statement gives the meaning of a local minimum at \[c\]?
