Advertisements
Advertisements
प्रश्न
Find the points at which the function f given by f (x) = (x – 2)4 (x + 1)3 has
- local maxima
- local minima
- point of inflexion
Advertisements
उत्तर
Here, f (x) = (x - 2)4 (x + 1)4
∴ f'(x) = (x – 2)4 · 3(x + 1)2 + (x + 1)3 · 4(x - 2)3
= (x – 2)3 (x + 1)2 [3(x – 2) + 4(x + 1)]
= (x – 2)3 (x + 1)2 [3x – 6 + 4x + 4]
= (x – 2)3 (x + 1)2 (7x – 2)
= 7(x - 2)3 (x + 1)2 `(x - 2/7)`
For maximum/minimum, 1(x) = 0
⇒ 7(x - 2)3 + (x + 1)2 `(x - 2/7)` = 0
∴ x = 2, -1, `2/7`
(i) When x = 2,
x is near 2 and to the left of 2 then, f(x) = (-)(+)(+) = -ve
x is near 2 and to the right of 2 then, f(x) = (+)(+)(+) = + ve
∴ The sign of f(x) changes from negative to positive as x passes through x = -2.
⇒ f is minimum at x = 2.
(ii) At, x = -1
For values of x near -1 and less than 1,
f'(x) = (-)(+)(-) = + ve
For values of x near to -1 and greater than -1,
f(x) = (-)(+)(-) = + ve
does not change its sign while passing through the point x = -1.
⇒ Thus, x = -1 is a point of inflexion
(iii) At, x = `2/7` = 0.28
On placing the value of x less than `2/7` near `2/7`,
f'(x) = (-)(+)(-) = + ve
Keeping the value of x close to `2/7` and greater than `2/7`,
f'(x) = (-)(+)(-) = -ve
⇒ At x = `2/7`, (x) changes from positive to negative.
As x passes through, x = `2/7`.
Thus it is minimum at x = 2, inflexion at x = -1 and a maximum at x = `2/7`.
APPEARS IN
संबंधित प्रश्न
Find the maximum and minimum value, if any, of the following function given by f(x) = 9x2 + 12x + 2
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:
g(x) = x3 − 3x
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
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 two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?
Show that a cylinder of a given volume, which is open at the top, has minimum total surface area when its height is equal to the radius of its base.
Find the maximum and minimum of the following functions : f(x) = `x^2 + (16)/x^2`
Divide the number 20 into two parts such that sum of their squares is minimum.
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.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.
Choose the correct option from the given alternatives :
If f(x) = `(x^2 - 1)/(x^2 + 1)`, for every real x, then the minimum value of f is ______.
State whether the following statement is True or False:
An absolute maximum must occur at a critical point or at an end point.
Examine the function for maxima and minima f(x) = x3 - 9x2 + 24x
The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is ______.
If f(x) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.
Twenty meters of wire is available for fencing off a flowerbed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is ______
The maximum value of function x3 - 15x2 + 72x + 19 in the interval [1, 10] is ______.
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.
If the sum of the lengths 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 `pi/3`
Find all the points of local maxima and local minima of the function f(x) = (x - 1)3 (x + 1)2
Find the local minimum value of the function f(x) `= "sin"^4" x + cos"^4 "x", 0 < "x" < pi/2`
The function f(x) = x5 - 5x4 + 5x3 - 1 has ____________.
The range of a ∈ R for which the function f(x) = `(4a - 3)(x + log_e5) + 2(a - 7)cot(x/2)sin^2(x/2), x ≠ 2nπ, n∈N` has critical points, is ______.
The lateral edge of a regular rectangular pyramid is 'a' cm long. The lateral edge makes an angle a. with the plane of the base. The value of a for which the volume of the pyramid is greatest, is ______.
Read the following passage:
Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore.
|
Based on the above information, answer the following questions:
- If the radius of cylinder is r cm and height is h cm, then write the volume V of cylinder in terms of radius r. (1)
- Find `(dV)/(dr)`. (1)
- (a) Find the radius of cylinder when its volume is maximum. (2)
OR
(b) For maximum volume, h > r. State true or false and justify. (2)
If f(x) = `1/(4x^2 + 2x + 1); x ∈ R`, then find the maximum value of f(x).
A box with a square base is to have an open top. The surface area of box is 147 sq. cm. What should be its dimensions in order that the volume is largest?
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
The shortest distance between the line y - x = 1and the curve x = y2 is
The absolute maximum value of the function f(x) = 2x3 − 3x2 − 36x + 9 defined on [−3, 3] is ______.
For a function defined on an interval \[I\], which condition means that \[f\] has a minimum value at \[c\in I\]?
Which statement gives the meaning of a local maximum at \[c\]?
What is an absolute minimum?
Assume \[f'(c)=0\] and the second derivative exists at \[c\]. Which condition gives a local maximum?
For \[f(x)=3x^4+4x^3-12x^2+12\], what conclusion follows at \[x=1\]?
