Advertisements
Advertisements
Question
Find the maximum value of 2x3 − 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [−3, −1].
Advertisements
Solution
f(x) = 2x3 - 24x + 107, interval [1, 3]
f'(x) = 6x2 - 24
For the highest and lowest values, f‘(x) = 0
⇒ 6x2 - 24 = 0
⇒ 6x2 = 24
⇒ x2 = 4
⇒ x = ± 2
Putting the values of x in f(x) = 2x3 - 24x + 107 for the interval [1, 3],
At, x = 1 f(1) = 2(1)3 - 24 (1) + 107 = 2 - 24 + 107 = 85
At, x = 3 f (3) = 2(3)3 - 24 (3) + 107 = 54 - 72 + 107 = 89
At, x = 2 f(2) = 2(2)3 - 24(2) + 107 = 16 - 48 + 107 = 75
Thus, the maximum value of f(x) = 89,
At x = 3, for the interval [-3,-1] we find the value of f(x) at x = - 3, - 2, - 1.
At, x = – 3 f(-3) = 2(-3)3 - 24 (-3) + 107 = - 54 + 72 + 107 = - 54 + 179 = 125
At, x = – 1 f(-1) = 2 (-1)3 - 24 (-1) + 107 = -2 +24 + 107 = 129
At, x = - 2 f(-2) = 2(-2)3 - 24 (-2) + 107 = -16 + 48 +107 = 139
Thus, the maximum value of f(x) = 139 at x = -2.
APPEARS IN
RELATED QUESTIONS
Find the maximum and minimum value, if any, of the function given by f(x) = |x + 2| − 1.
Find the maximum and minimum value, if any, of the following function given by f(x) = |sin 4x + 3|
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) = 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:
f(x) = sinx − cos x, 0 < x < 2π
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `tan^(-1) sqrt(2)`
A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi-circular ends is \[\pi : (\pi + 2)\].
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.
Find the maximum and minimum of the following functions : y = 5x3 + 2x2 – 3x.
Find the maximum and minimum of the following functions : f(x) = x log x
A wire of length 36 metres is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.
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.
If x + y = 3 show that the maximum value of x2y is 4.
Examine the function for maxima and minima f(x) = x3 - 9x2 + 24x
Find the local maximum and local minimum value of f(x) = x3 − 3x2 − 24x + 5
By completing the following activity, examine the function f(x) = x3 – 9x2 + 24x for maxima and minima
Solution: f(x) = x3 – 9x2 + 24x
∴ f'(x) = `square`
∴ f''(x) = `square`
For extreme values, f'(x) = 0, we get
x = `square` or `square`
∴ f''`(square)` = – 6 < 0
∴ f(x) is maximum at x = 2.
∴ Maximum value = `square`
∴ f''`(square)` = 6 > 0
∴ f(x) is maximum at x = 4.
∴ Minimum value = `square`
If f(x) = px5 + qx4 + 5x3 - 10 has local maximum and minimum at x = 1 and x = 3 respectively then (p, q) = ______.
If z = ax + by; a, b > 0 subject to x ≤ 2, y ≤ 2, x + y ≥ 3, x ≥ 0, y ≥ 0 has minimum value at (2, 1) only, then ______.
The sum of two non-zero numbers is 6. The minimum value of the sum of their reciprocals is ______.
The smallest value of the polynomial x3 – 18x2 + 96x in [0, 9] is ______.
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`
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`
Find both the maximum and minimum values respectively of 3x4 - 8x3 + 12x2 - 48x + 1 on the interval [1, 4].
Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a.
The maximum value of the function f(x) = `logx/x` is ______.
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 ______.
The greatest value of the function f(x) = `tan^-1x - 1/2logx` in `[1/sqrt(3), sqrt(3)]` 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 ______.
The maximum value of z = 6x + 8y subject to constraints 2x + y ≤ 30, x + 2y ≤ 24 and x ≥ 0, y ≥ 0 is ______.
Read the following passage:
Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore.
|
Based on the above information, answer the following questions:
- If the radius of cylinder is r cm and height is h cm, then write the volume V of cylinder in terms of radius r. (1)
- Find `(dV)/(dr)`. (1)
- (a) Find the radius of cylinder when its volume is maximum. (2)
OR
(b) For maximum volume, h > r. State true or false and justify. (2)
Find the maximum profit that a company can make, if the profit function is given by P(x) = 72 + 42x – x2, where x is the number of units and P is the profit in rupees.
If x + y = 8, then the maximum value of x2y is ______.
Find the point on the curve y2 = 4x, which is nearest to the point (2, 1).
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.
