Advertisements
Advertisements
Question
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.
Advertisements
Solution
We have f(x) = x5 – 5x4 + 5x3 – 1
⇒ f '(x) = 5x4 – 20x3 + 15x2
For local maxima and local minima, f '(x) = 0
⇒ 5x4 – 20x3 + 15x2 = 0
⇒ 5x2(x2 – 4x + 3) = 0
⇒ 5x2(x2 – 3x – x + 3) = 0
⇒ x2(x – 3)(x – 1) = 0
∴ x = 0, x = 1 and x = 3
Now f '(x) = 20x3 – 60x2 + 30x
⇒ `"f''"(x)_("at" x = 0)` = 20(0)3 – 60(0)2 + 30(0) = 0
Which is neither maxima nor minima.
∴ f (x) has the point of inflection at x = 0
`"f''"(x)_("at" x = 1)` = 20(1)3 – 60(1)2 + 30(1)
= 20 – 60 + 30
= –10 < 0 Maxima
`"f''"(x)_("at" x = 2)` = 20(3)3 – 60(3)2 + 30(3)
= 540 – 540 + 90
= 90 > 0 Minima
The maximum value of the function at x = 1
f (x) = (1)5 – 5(1)4 + 5(1)3 – 1
= 1 – 5 + 5 – 1
= 0
The minimum value at x = 3 is
f (x) = (3)5 – 5(3)4 + 5(3)3 – 1
= 243 – 405 + 135 – 1
= 378 – 406
= – 28
Hence, the function has its maxima at x = 1 and the maximum value = 0 and it has minimum value at x = 3 and its minimum value is – 28.
APPEARS IN
RELATED QUESTIONS
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`. Also find maximum volume in terms of volume of the sphere
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 function given by f(x) = |x + 2| − 1.
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:
f(x) = ex
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
`f(x) =x^3, x in [-2,2]`
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 two numbers whose sum is 24 and whose product is as large as possible.
Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is `8/27` of the volume of the sphere.
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
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 maximum and minimum of the following functions : f(x) = x log x
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.
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)`.
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?
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`
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 ______
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.
If x is real, the minimum value of x2 – 8x + 17 is ______.
The smallest value of the polynomial x3 – 18x2 + 96x in [0, 9] is ______.
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`
If y `= "ax - b"/(("x" - 1)("x" - 4))` has a turning point P(2, -1), then find the value of a and b respectively.
Find both the maximum and minimum values respectively of 3x4 - 8x3 + 12x2 - 48x + 1 on the interval [1, 4].
The function f(x) = x5 - 5x4 + 5x3 - 1 has ____________.
The area of a right-angled triangle of the given hypotenuse is maximum when the triangle 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 y = alog|x| + bx2 + x has its extremum values at x = –1 and x = 2, then ______.
The function g(x) = `(f(x))/x`, x ≠ 0 has an extreme value when ______.
Let f(x) = (x – a)ng(x) , where g(n)(a) ≠ 0; n = 0, 1, 2, 3.... then ______.
The maximum value of z = 6x + 8y subject to constraints 2x + y ≤ 30, x + 2y ≤ 24 and x ≥ 0, y ≥ 0 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 ______.
Check whether the function f : R `rightarrow` R defined by f(x) = x3 + x, has any critical point/s or not ? If yes, then find the point/s.
If x + y = 8, then the maximum value of x2y is ______.
Determine the minimum value of the function.
f(x) = 2x3 – 21x2 + 36x – 20
20 is divided into two parts so that the product of the cube of one part and the square of the other part is maximum, then these two parts are
