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.
Given: function f (x) = x5 – 5x4 + 5x3 – 1
To find: the points of local maxima, local minima and the points of inflection of f(x) and also to find the corresponding local maximum and local minimum values.
Explanation: given f (x) = x5 – 5x4 + 5x3 – 1
We will find the first derivative of f(x), i.e.,
Applying the sum rule of differentiation and taking out the constant term, we get
f' (x) = 5x4-5.4x3+5.3x2-0
⇒ f' (x) = 5x4-20x3+15x2
Now to find the critical point we need to equate the first derivative to 0, i.e.,
f'(x) = 0
⇒ 5x4-20x3+15x2 = 0
⇒ 5x2(x2-4x+3) = 0
⇒ 5x2(x2-4x+3) = 0
Now splitting the middle term, we get
⇒ 5x2(x2-3x-x+3) = 0
⇒ 5x2[ x(x-3)-1(x-3)] = 0
⇒ 5x2(x-1)(x-3) = 0
⇒ 5x2 = 0 or (x-1) = 0 or (x-3) = 0
⇒ x = 0 or x = 1 or x = 3
Now we will find the corresponding y value by substituting the different values of x in given function f(x) = x5 – 5x4 + 5x3 – 1,
When x = 0, f(0) = 05 – 5(0)4 + 5(0)3 – 1 = -1
Hence the point is (0,-1)
When x = 1, f(1) = 15 – 5(1)4 + 5(1)3 – 1 = 1-5+5-1 = 0
Hence the point is (0,0)
When x = 3, f(3) = 35 – 5(3)4 + 5(3)3 – 1 = 243-405+135-1 = -28
Hence the point is (0,-28)
Therefore, we see that
At x = 3, y has minimum value = -28. Hence x = 3 is point of local minima.
At x = 1, y has maximum value = 0. Hence x = 1 is point of local minima.
And at x = 0, y has neither maximum nor minimum value, hence this is point of inflection.
Rate this question :
If the sum of the hypotenuse and a side of a right-angled triangle is given. Show that the area of the triangle is maximum when the angle between them is
A manufacturer can sell x items at a price of Rs. each. The cost price of x items is Rs. Find the number of items he should sell to earn maximum profit.Mathematics - Board Papers
A metal box with a square base and vertical sides is to contain 1024 cm3. The material for the top and bottom costs ` 5 per cm2 and the material for the sides costs ` 2.50 per cm2. Find the least cost of the box.Mathematics - Board Papers
Show that among all positive numbers x and y with x2 + y2 = r2, the sum x + y is largest when x = y = .RD Sharma - Volume 1
Find the local maxima and local minima, of the function f(x) sin x – cos x, 0< x < 2π .Also, find the local maximum and local minimum values.Mathematics - Board Papers
Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface area is cot-1√2.Mathematics - Board Papers
Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone.
An open box with a square base is to be made out of a given quantity of cardboard of area c2 square units. Show that the maximum volume of the box iscubic units.Mathematics - Board Papers
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed isMathematics - Board Papers