Advertisements
Advertisements
Question
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:
f(x) = x3 − 6x2 + 9x + 15
Advertisements
Solution
Given function, f(x) = x3 - 6x2 + 9x + 15
`therefore` f'(x) = 3x2 - 12x + 9
= 3 (x2 - 4x + 3)
= 3 (x - 1)(x - 3)
=> x = 1 or x = 3
`therefore` x can have lowest or highest value at x = 1 or x = 3.
f''(x) = 3(2x - 4) = 6x - 12
At, x = 1, f''(x) = 6 × 1 - 12 = - 6 (Negative)
∴ The value of the function at x = 1 is a local maximum.
Maximum value = f(1) = (1)3 - 6(1)2 + 9(1) + 15
= 1 - 6 + 9 + 15
= 19
At x = 3, f'' = 6 × 3 - 12 = 6 positive
∴ f(x) has a local minimum at x = 3.
Minimum value = f(3) = (3)3 - 6(3)2 + 9(3) + 15
= 27 - 54 + 27 + 15
= 15
APPEARS IN
RELATED QUESTIONS
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 h(x) = sin(2x) + 5.
Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
`f(x) = xsqrt(1-x), x > 0`
Prove that the following function do not have maxima or minima:
h(x) = x3 + x2 + x + 1
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 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]`
Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.
Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
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?
A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `tan^(-1) sqrt(2)`
Find the maximum and minimum of the following functions : f(x) = x log x
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?
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)`.
If x + y = 3 show that the maximum value of x2y is 4.
The function f(x) = x log x is minimum at x = ______.
If f(x) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.
If R is the circum radius of Δ ABC, then A(Δ ABC) = ______.
If f(x) = `x + 1/x, x ne 0`, then local maximum and x minimum values of function f are respectively.
Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its sides. Also, find the maximum volume.
Find the points of local maxima and local minima respectively for the function f(x) = sin 2x - x, where `-pi/2 le "x" le pi/2`
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 ____________.
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)
Range of projectile will be maximum when angle of projectile is
The maximum value of `[x(x - 1) + 1]^(2/3), 0 ≤ x ≤ 1` is
The maximum value of the function f(x) = `logx/x` is ______.
Divide 20 into two ports, so that their product is maximum.
If S1 and S2 are respectively the sets of local minimum and local maximum points of the function. f(x) = 9x4 + 12x3 – 36x2 + 25, x ∈ R, then ______.
The point in the interval [0, 2π], where f(x) = ex sin x has maximum slope, is ______.
Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.
A metal wire of 36 cm long is bent to form a rectangle. Find its dimensions when its area is maximum.
If x + y = 8, then the maximum value of x2y is ______.
Mrs. Roy designs a window in her son’s study room so that the room gets maximum sunlight. She designs the window in the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 m, find the dimensions of the window that will admit maximum sunlight into the room.
