Advertisements
Advertisements
Question
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.
Advertisements
Solution
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of Local minima.
APPEARS IN
RELATED QUESTIONS
Find the maximum and minimum value, if any, of the following function given by f(x) = −(x − 1)2 + 10
Find the maximum and minimum value, if any, of the following function given by f(x) = |sin 4x + 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:
f(x) = x2
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:
`h(x) = sinx + cosx, 0 < x < pi/2`
What is the maximum value of the function sin x + cos x?
Find two numbers whose sum is 24 and whose product is as large as possible.
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)`
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
For all real values of x, the minimum value of `(1 - x + x^2)/(1+x+x^2)` is ______.
Find the maximum area of an isosceles triangle inscribed in the ellipse `x^2/ a^2 + y^2/b^2 = 1` with its vertex at one end of the major axis.
A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening
Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
A metal box with a square base and vertical sides is to contain 1024 cm3. The material for the top and bottom costs Rs 5 per cm2 and the material for the sides costs Rs 2.50 per cm2. Find the least cost of the box
An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when the depth of the tank is half of its width. If the cost is to be borne by nearby settled lower-income families, for whom water will be provided, what kind of value is hidden in this question?
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.
The profit function P(x) of a firm, selling x items per day is given by P(x) = (150 – x)x – 1625 . Find the number of items the firm should manufacture to get maximum profit. Find the maximum profit.
Solve the following : Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is `(2"R")/sqrt(3)`. Also, find the maximum volume.
Determine the maximum and minimum value of the following function.
f(x) = x log x
If f(x) = x.log.x then its maximum value is ______.
A wire of length 120 cm is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum
The maximum and minimum values for the function f(x) = 4x3 - 6x2 on [-1, 2] are ______
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 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 smallest value of the polynomial x3 – 18x2 + 96x in [0, 9] is ______.
If y = x3 + x2 + x + 1, then y ____________.
The coordinates of the point on the parabola y2 = 8x which is at minimum distance from the circle x2 + (y + 6)2 = 1 are ____________.
The function `"f"("x") = "x" + 4/"x"` has ____________.
Let f(x) = 1 + 2x2 + 22x4 + …… + 210x20. Then f (x) has ____________.
A ball is thrown upward at a speed of 28 meter per second. What is the speed of ball one second before reaching maximum height? (Given that g= 10 meter per second2)
Divide 20 into two ports, so that their product is maximum.
Read the following passage and answer the questions given below.
|
In an elliptical sport field the authority wants to design a rectangular soccer field with the maximum possible area. The sport field is given by the graph of `x^2/a^2 + y^2/b^2` = 1. |
- If the length and the breadth of the rectangular field be 2x and 2y respectively, then find the area function in terms of x.
- Find the critical point of the function.
- Use First derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
OR
Use Second Derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is k (meter), then `(4/π + 1)`k 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 ______.
The maximum value of z = 6x + 8y subject to constraints 2x + y ≤ 30, x + 2y ≤ 24 and x ≥ 0, y ≥ 0 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"`
A right circular cylinder is to be made so that the sum of the radius and height is 6 metres. Find the maximum volume of the cylinder.
