Advertisements
Advertisements
Question
For all real values of x, the minimum value of `(1 - x + x^2)/(1+x+x^2)` is ______.
Options
0
1
3
`1/3`
Advertisements
Solution
For all real values of x, the minimum value of `(1 - x + x^2)/(1+x+x^2)` is `underline(1/3)`.
Explanation:
Let, y = `(1 - x + x^2)/(1 + x = x^2)`
Differentiating both sides with respect to x,
`dy/dx = ((-1 + 2x)(1 + x + x^2) - (1 - x + x^2)(1 + 2x))/((1 + x + x^2)^2)`
`= ((- 1 - x + x^2) + (2x + 2x^2 + 2x^3) - [(1 - x + x^2 + 2x - 2x^2 = 2x^3)])/((1 + x + x^2)^2)`
`= (-1 - x - x^2 + 2x + 2x^2 + 2x^3 - 1 + x - x^2 - 2x + 2x^2 - 2x^3)/((1 + x + x^2)^2)`
`= (-2 + 2x^2)/((1 + x + x^2)^2)`
`= (2 (x^2 - 1))/((1 + x + x^2)^2)`
`= (2(x - 1)(x + 1))/((1 + x + x^2)^2)`
For highest and lowest value, `dy/dx = 0`
`therefore (2(x - 1)(x + 1))/((1 + x + x^2)^2) = 0/1`
`=>` (x - 1)(x + 1) = 0
`therefore` x = 1, -1
The sign of `dy/dx` at x = 1 changes from negative to positive when the point moves through x = 1.
`therefore` y is minimum at the point x = 1.
Minimum value, f(1) `= (1 - 1 + 1)/(1 + 1 + 1) = 1/3`
APPEARS IN
RELATED QUESTIONS
Find the maximum and minimum value, if any, of the following function given by g(x) = − |x + 1| + 3.
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 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 their sum is 35 and the product x2y5 is a maximum.
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)`
Show that semi-vertical angle of right circular cone of given surface area and maximum volume is `Sin^(-1) (1/3).`
A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle.
Show that the minimum length of the hypotenuse is `(a^(2/3) + b^(2/3))^(3/2).`
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 point on the straight line 2x+3y = 6, which is closest to the origin.
A ball is thrown in the air. Its height at any time t is given by h = 3 + 14t – 5t2. Find the maximum height it can reach.
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?
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)`.
A metal wire of 36 cm length is bent to form a rectangle. Find its dimensions when its area is maximum.
The total cost of producing x units is ₹ (x2 + 60x + 50) and the price is ₹ (180 − x) per unit. For what units is the profit maximum?
A rectangular sheet of paper has it area 24 sq. Meters. The margin at the top and the bottom are 75 cm each and the sides 50 cm each. What are the dimensions of the paper if the area of the printed space is maximum?
The sum of the surface areas of a rectangular parallelopiped with sides x, 2x and `x/3` and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of the sphere. Also find the minimum value of the sum of their volumes.
The maximum value of `(1/x)^x` is ______.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.
The coordinates of the point on the parabola y2 = 8x which is at minimum distance from the circle x2 + (y + 6)2 = 1 are ____________.
Let f(x) = 1 + 2x2 + 22x4 + …… + 210x20. Then f (x) has ____________.
Divide 20 into two ports, so that their product is maximum.
If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to ______.
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 ______.
Let x and y be real numbers satisfying the equation x2 – 4x + y2 + 3 = 0. If the maximum and minimum values of x2 + y2 are a and b respectively. Then the numerical value of a – b is ______.
The set of values of p for which the points of extremum of the function f(x) = x3 – 3px2 + 3(p2 – 1)x + 1 lie in the interval (–2, 4), is ______.
Let f(x) = |(x – 1)(x2 – 2x – 3)| + x – 3, x ∈ R. If m and M are respectively the number of points of local minimum and local maximum of f in the interval (0, 4), then m + M 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.
If f(x) = `1/(4x^2 + 2x + 1); x ∈ R`, then find the maximum value of f(x).
A metal wire of 36 cm long is bent to form a rectangle. Find its dimensions when its area is maximum.
Find the maximum and the minimum values of the function f(x) = x2ex.
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) `= x sqrt(1 - x), 0 < x < 1`
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 absolute maximum value of the function f(x) = 2x3 − 3x2 − 36x + 9 defined on [−3, 3] is ______.
Which statement gives the meaning of a local maximum at \[c\]?
Why must the left endpoint \[a\] and right endpoint \[b\] be checked for absolute extrema on \[[a,b]\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], which is the correct factorization of \[f'(x)\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], what conclusion follows at \[x=0\]?
