Advertisements
Advertisements
प्रश्न
If f(x) = `1/(4x^2 + 2x + 1); x ∈ R`, then find the maximum value of f(x).
उत्तर १
f(x) = `1/(4x^2 + 2x + 1)`,
Let g(x) = 4x2 + 2x + 1
= `4(x^2 + 2x 1/4 + 1/16) + 3/4`
= `4(x + 1/4)^2 + 3/4 ≥ 3/4`
∴ Maximum value of f(x) = `4/3`.
उत्तर २
f(x) = `1/(4x^2 + 2x + 1)`,
Let f(x) = 4x2 + 2x + 1
`\implies d/dx(g(x))` = g(x) = 8x + 2 and g(x) = 0 at x = `-1/4` also `d^2/(dx^2)(g(x))` = g"(x) = 8 > 0
`\implies` g(x) is minimum when x = `-1/4` so, f(x) is maximum at x = `-1/4`
∴ Maximum value of f(x) = `f(-1/4) = 1/(4(-1/4)^2 + 2(-1/4) + 1) = 4/3`.
उत्तर ३
f(x) = `1/(4x^2 + 2x + 1)`
On differentiating w.r.t x, we get
f'(x) = `(-(8x + 2))/(4x^2 + 2x + 1)^2` ....(i)
For maxima or minima, we put
f'(x) = 0
`\implies` 8x + 2 = 0
`\implies` x = `-1/4`.
Again, differentiating equation (i) w.r.t. x, we get
f"(x) = `-{((4x^2 + 2x + 1)^2 (8) - (8x + 2)2 xx (4x^2 + 2x + 1)(8x + 2))/(4x^2 + 2x + 1)^4}`
At x = `-1/4, f^('')(-1/4) < 0`
f(x) is maximum at x = `-1/4`.
∴ Maximum value of f(x) is `f(-1/4) = 1/(4(-1/4)^2 + 2(-1/4) + 1) = 4/3`.
उत्तर ४
f(x) = `1/(4x^2 + 2x + 1)`
On differentiating w.r.t x,we get
f'(x) = `(-(8x + 2))/(4x^2 + 2x + 1)^2` ....(i)
For maxima or minima, we put
f'(x) = 0
`\implies` 8x + 2 = 0
`\implies` x = `-1/4`.
When `x ∈ (-h - 1/4, -1/4)`, where 'h' is infinitesimally small positive quantity.
4x < – 1
`\implies` 8x < –2
`\implies` 8x + 2 < 0
`\implies` –(8x + 2) > 0 and (4x2 + 2x + 1)2 > 0
`\implies` f'(x) > 0
and when `x ∈ (-1/4, -1/4 + h)`, 4x > –1
`\implies` 8x > – 2
`\implies` 8x + 2 > 0
`\implies` – (8x + 2) < 0
and (4x2 + 2x + 1)2 > 0
`\implies` f'(x) < 0.
This shows that x = `-1/4` is the point of local maxima.
∴ Maximum value of f(x) is `f(-1/4) = 1/(4(-1/4)^2 + 2(-1/4) + 1) = 4/3`.
APPEARS IN
संबंधित प्रश्न
Show that the height of the cylinder of maximum volume, that can be inscribed in a sphere of radius R is `(2R)/sqrt3.` Also, find the maximum volume.
Find the maximum and minimum value, if any, of the following function given by h(x) = x + 1, x ∈ (−1, 1)
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) = x/2 + 2/x, x > 0`
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) = 1/(x^2 + 2)`
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 two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
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?
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
Find the maximum and minimum of the following functions : f(x) = 2x3 – 21x2 + 36x – 20
An open cylindrical tank whose base is a circle is to be constructed of metal sheet so as to contain a volume of `pia^3`cu cm of water. Find the dimensions so that the quantity of the metal sheet required is minimum.
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?
The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is ______.
If R is the circum radius of Δ ABC, then A(Δ ABC) = ______.
Show that the function f(x) = 4x3 – 18x2 + 27x – 7 has neither maxima nor minima.
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 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.
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.
The maximum value of `["x"("x" − 1) + 1]^(1/3)`, 0 ≤ x ≤ 1 is:
Find the local minimum value of the function f(x) `= "sin"^4" x + cos"^4 "x", 0 < "x" < pi/2`
Let f(x) = 1 + 2x2 + 22x4 + …… + 210x20. Then f (x) has ____________.
Range of projectile will be maximum when angle of projectile is
For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`
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 ______.
If y = alog|x| + bx2 + x has its extremum values at x = –1 and x = 2, then ______.
A cone of maximum volume is inscribed in a given sphere. Then the ratio of the height of the cone to the diameter of the sphere is ______.
A rectangle with one side lying along the x-axis is to be inscribed in the closed region of the xy plane bounded by the lines y = 0, y = 3x and y = 30 – 2x. The largest area of such a rectangle is ______.
The maximum value of f(x) = `logx/x (x ≠ 0, x ≠ 1)` is ______.
The rectangle has area of 50 cm2. Complete the following activity to find its dimensions for least perimeter.
Solution: Let x cm and y cm be the length and breadth of a rectangle.
Then its area is xy = 50
∴ `y =50/x`
Perimeter of rectangle `=2(x+y)=2(x+50/x)`
Let f(x) `=2(x+50/x)`
Then f'(x) = `square` and f''(x) = `square`
Now,f'(x) = 0, if x = `square`
But x is not negative.
∴ `x = root(5)(2) "and" f^('')(root(5)(2))=square>0`
∴ by the second derivative test f is minimum at x = `root(5)(2)`
When x = `root(5)(2),y=50/root(5)(2)=root(5)(2)`
∴ `x=root(5)(2) "cm" , y = root(5)(2) "cm"`
Hence, rectangle is a square of side `root(5)(2) "cm"`
Find the maximum and the minimum values of the function f(x) = x2ex.
