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:
f(x) = x3 − 6x2 + 9x + 15
Advertisements
उत्तर
Given function, f(x) = x3 - 6x2 + 9x + 15
`therefore` f'(x) = 3x2 - 12x + 9
= 3 (x2 - 4x + 3)
= 3 (x - 1)(x - 3)
=> x = 1 or x = 3
`therefore` x can have lowest or highest value at x = 1 or x = 3.
f''(x) = 3(2x - 4) = 6x - 12
At, x = 1, f''(x) = 6 × 1 - 12 = - 6 (Negative)
∴ The value of the function at x = 1 is a local maximum.
Maximum value = f(1) = (1)3 - 6(1)2 + 9(1) + 15
= 1 - 6 + 9 + 15
= 19
At x = 3, f'' = 6 × 3 - 12 = 6 positive
∴ f(x) has a local minimum at x = 3.
Minimum value = f(3) = (3)3 - 6(3)2 + 9(3) + 15
= 27 - 54 + 27 + 15
= 15
APPEARS IN
संबंधित प्रश्न
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.
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/3`. Also find maximum volume in terms of volume of the sphere
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 functions. Find also the local maximum and the local minimum values, as the case may be:
`f(x) = xsqrt(1-x), x > 0`
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]
At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?
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?
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 maximum value of `[x(x −1) +1]^(1/3)` , 0 ≤ x ≤ 1 is ______.
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 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`
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 : 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)`.
Find the local maximum and local minimum value of f(x) = x3 − 3x2 − 24x + 5
If R is the circum radius of Δ ABC, then A(Δ ABC) = ______.
The minimum value of the function f(x) = 13 - 14x + 9x2 is ______
A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs 300/- per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Re 1/- one subscriber will discontinue the service. Find what increase will bring maximum profit?
If x is real, the minimum value of x2 – 8x + 17 is ______.
The smallest value of the polynomial x3 – 18x2 + 96x in [0, 9] is ______.
The maximum value of `(1/x)^x` is ______.
Find the points of local maxima and local minima respectively for the function f(x) = sin 2x - x, where `-pi/2 le "x" le pi/2`
If y `= "ax - b"/(("x" - 1)("x" - 4))` has a turning point P(2, -1), then find the value of a and b respectively.
Find the maximum profit that a company can make, if the profit function is given by P(x) = 41 + 24x – 18x2.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.
The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is ____________.
Let f: R → R be a function defined by f(x) = (x – 3)n1(x – 5)n2, n1, n2 ∈ N. Then, which of the following is NOT true?
Let P(h, k) be a point on the curve y = x2 + 7x + 2, nearest to the line, y = 3x – 3. Then the equation of the normal to the curve at P is ______.
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 y = alog|x| + bx2 + x has its extremum values at x = –1 and x = 2, then ______.
If the point (1, 3) serves as the point of inflection of the curve y = ax3 + bx2 then the value of 'a ' and 'b' are ______.
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 ______.
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 ______.
The maximum distance from origin of a point on the curve x = `a sin t - b sin((at)/b)`, y = `a cos t - b cos((at)/b)`, both a, b > 0 is ______.
The volume of the greatest cylinder which can be inscribed in a cone of height 30 cm and semi-vertical angle 30° is ______.
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.
Divide the number 100 into two parts so that the sum of their squares is minimum.
Sumit has bought a closed cylindrical dustbin. The radius of the dustbin is ‘r' cm and height is 'h’ cm. It has a volume of 20π cm3.
- Express ‘h’ in terms of ‘r’, using the given volume.
- Prove that the total surface area of the dustbin is `2πr^2 + (40π)/r`
- Sumit wants to paint the dustbin. The cost of painting the base and top of the dustbin is ₹ 2 per cm2 and the cost of painting the curved side is ₹ 25 per cm2. Find the total cost in terms of ‘r’, for painting the outer surface of the dustbin including the base and top.
- Calculate the minimum cost for painting the dustbin.
