Advertisements
Advertisements
प्रश्न
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`
Advertisements
उत्तर
Given function g(x) = `x/2 + 2/x, x > 0`
`g'(x) = 1/2 - 2/(x^2), x > 0`
`= (x^2 - 4)/(2x^2)`
`= ((x - 2)(x + 2))/(2x^2)`
g'(x) = 0 ⇒ (x - 2)(x + 2) = 0
∴ x = -2, 2
∴ Critical points are -2 and 2, but x > 0
∴ Critical point = 2
Now, g''(x) = 0 `- (3 xx 2)/x^3 = 6/x^3`
At x = 2, g''(x) = `6/2^3 = 3/4` (positive)
g'' (x) = -2 (-2) x-3 , x > 0 and
g'' (2) = 4(2)-3 = `4/8 = 1/2 > 0`
∴ At x = 2, the value of g(x) is minimum
And minimum value = g(2) = `2/2 + 2/2 = 2`
APPEARS IN
संबंधित प्रश्न
Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]
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.
A telephone company in a town has 5000 subscribers on its list and collects fixed rent charges of Rs.3,000 per year from each subscriber. The company proposes to increase annual rent and it is believed that for every increase of one rupee in the rent, one subscriber will be discontinued. Find what increased annual rent will bring the maximum annual income to the company.
Find the maximum and minimum value, if any, of the following function given by f(x) = 9x2 + 12x + 2
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:
f(x) = x3 − 6x2 + 9x + 15
Prove that the following function do not have maxima or minima:
g(x) = logx
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 profit that a company can make, if the profit function is given by p(x) = 41 − 72x − 18x2.
Find both the maximum value and the minimum value of 3x4 − 8x3 + 12x2 − 48x + 25 on the interval [0, 3].
Find the maximum and minimum values of x + sin 2x on [0, 2π].
Show that semi-vertical angle of right circular cone of given surface area and maximum volume is `Sin^(-1) (1/3).`
Find the absolute maximum and minimum values of the function f given by f (x) = cos2 x + sin x, x ∈ [0, π].
Find the point on the straight line 2x+3y = 6, which is closest to the origin.
Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.
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 : Show that a closed right circular cylinder of given surface area has maximum volume if its height equals the diameter of its base.
Determine the maximum and minimum value of the following function.
f(x) = `x^2 + 16/x`
State whether the following statement is True or False:
An absolute maximum must occur at a critical point or at an end point.
Find the local maximum and local minimum value of f(x) = x3 − 3x2 − 24x + 5
The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is ______.
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.
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
The maximum value of `(1/x)^x` is ______.
The maximum value of `["x"("x" − 1) + 1]^(1/3)`, 0 ≤ x ≤ 1 is:
Find both the maximum and minimum values respectively of 3x4 - 8x3 + 12x2 - 48x + 1 on the interval [1, 4].
Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.
Find the area of the largest isosceles triangle having a perimeter of 18 meters.
The function `"f"("x") = "x" + 4/"x"` has ____________.
The combined resistance R of two resistors R1 and R2 (R1, R2 > 0) is given by `1/"R" = 1/"R"_1 + 1/"R"_2`. If R1 + R2 = C (a constant), then maximum resistance R is obtained if ____________.
Read the following passage and answer the questions given below.
|
|
- Is the function differentiable in the interval (0, 12)? Justify your answer.
- If 6 is the critical point of the function, then find the value of the constant m.
- Find the intervals in which the function is strictly increasing/strictly decreasing.
OR
Find the points of local maximum/local minimum, if any, in the interval (0, 12) as well as the points of absolute maximum/absolute minimum in the interval [0, 12]. Also, find the corresponding local maximum/local minimum and the absolute ‘maximum/absolute minimum values of the function.
A function f(x) is maximum at x = a when f'(a) > 0.
A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is ______.
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 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 ______.
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 ______.
Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.
Divide the number 100 into two parts so that the sum of their squares is minimum.
