Advertisements
Advertisements
Question
The function f(x) = 2x3 – 3x2 – 12x + 4, has ______.
Options
Two points of local maximum
Two points of local minimum
One maxima and one minima
No maxima or minima
Advertisements
Solution
The function f(x) = 2x3 – 3x2 – 12x + 4, has one maxima and one minima.
Explanation:
We have f(x) = 2x3 – 3x2 – 12x + 4
f'(x) = 6x2 – 6x – 12
For local maxima and local minima f'(x) = 0
∴ 6x2 – 6x – 12 = 0
⇒ x2 – x – 2 = 0
⇒ x2 – 2x + x – 2 = 0
⇒ x(x – 2) + 1(x – 2) = 0
⇒ (x + 1)(x – 2) = 0
x = –1, 2 are the points of local maxima and local minima
Now f'(x) = 12x – 6
`"f''"(x)_(x = -1)` = 12(–1) – 6
= – 12 – 6
= – 18 < 0, maxima
`"f''"(x)_(x = 2)` = 12(2) – 6
= 24 – 6
= 18 > 0 minima
So, the function is maximum at x = –1 and minimum at x = 2
APPEARS IN
RELATED QUESTIONS
Examine the maxima and minima of the function f(x) = 2x3 - 21x2 + 36x - 20 . Also, find the maximum and minimum values of f(x).
If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f(x).
An open box is to be made out of a piece of a square card board of sides 18 cms by cutting off equal squares from the comers and turning up the sides. Find the maximum volume of the box.
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 f(x) = (2x − 1)2 + 3.
Find the maximum and minimum value, if any, of the following function given by f(x) = |sin 4x + 3|
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) = (x −1)2 + 3, x ∈[−3, 1]
Find the maximum and minimum values of x + sin 2x on [0, 2π].
Find two numbers whose sum is 24 and whose product is as large as possible.
Find the absolute maximum and minimum values of the function f given by f (x) = cos2 x + sin x, x ∈ [0, π].
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.`
Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
The volume of a closed rectangular metal box with a square base is 4096 cm3. The cost of polishing the outer surface of the box is Rs. 4 per cm2. Find the dimensions of the box for the minimum cost of polishing it.
Find the point on the straight line 2x+3y = 6, which is closest to the origin.
Find the largest size of a rectangle that can be inscribed in a semicircle of radius 1 unit, so that two vertices lie on the diameter.
Show that among rectangles of given area, the square has least perimeter.
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.
Divide the number 20 into two parts such that their product is maximum.
A rod of 108 m long is bent to form a rectangle. Find it’s dimensions when it’s area is maximum.
Max value of z equals 3x + 2y subject to x + y ≤ 3, x ≤ 2, -2x + y ≤ 1, x ≥ 0, y ≥ 0 is ______
The function y = 1 + sin x is maximum, when x = ______
The minimum value of the function f(x) = 13 - 14x + 9x2 is ______
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`
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.
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.
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
Find the area of the largest isosceles triangle having a perimeter of 18 meters.
The coordinates of the point on the parabola y2 = 8x which is at minimum distance from the circle x2 + (y + 6)2 = 1 are ____________.
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 maximum distance from origin of a point on the curve x = `a sin t - b sin((at)/b)`, y = `a cos t - b cos((at)/b)`, both a, b > 0 is ______.
The minimum value of the function f(x) = xlogx is ______.
A rod AB of length 16 cm. rests between the wall AD and a smooth peg, 1 cm from the wall and makes an angle θ with the horizontal. The value of θ for which the height of G, the midpoint of the rod above the peg is minimum, 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 ______.
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.
Find the maximum and the minimum values of the function f(x) = x2ex.
If \[\mathrm{A}+\mathrm{B}=\frac{\pi}{2}\] then the maximum value of cosA.cosB is
A maximum or minimum value of \[f\] is called a(n) _____.
Assume \[f'(c)=0\] and the second derivative exists at \[c\]. Which condition gives a local minimum?
For a continuous function on a closed interval \[[a,b]\], which values must be compared to find absolute maxima and minima?
