Advertisements
Advertisements
Question
Find the maximum and minimum of the following functions : y = 5x3 + 2x2 – 3x.
Advertisements
Solution 1
y = 5x3 + 2x2 – 3x
∴ `dy/dx = d/dx(5x^3 + 2x^2 - 3x)`
= 5 × 3x2 + 2 × 2x – 3 × 1
= 15x2 + 4x – 3
and `(d^2y)/(dx^2) = d/dx(15x^2 + 4x - 3)`
= 15 × 2x + 4 x 1 – 0
= 30x + 4
`dy/dx` = 0 gives 15x2 + 4x – 3 = 0
∴ 15x2 + 9x – 5x – 3 = 0
∴ 3x(5x + 3) – 1(5x + 3) = 0
∴ (5x + 3)(3x – 1) = 0
∴ x = `-(3)/(5) or x = (1)/(3)`
∴ the roots of `dy/dx = 0` are `x_1 = -(3)/(5) and x_2 = (1)/(3)`.
Second Derivative Test:
(a) `((d^2y)/dx^2)_("at" x = - 3/5) = 30(-3/5) + 4` = – 14 < 0
∴ by the second derivative test, y is maximum at x = `-(3)/(5)` and maximum value of y at x = `-(3)/(5)`
= `5(-3/5)^3 + 2(-3/5)^2 - 3(-3/5)`
= `(-27)/(25) + (18)/(25) + (9)/(5)`
= `(36)/(25)`
(b) `((d^2y)/dx^2)_("at" x = 1/3) = 30(1/3) + 4` = 14 > 0
∴ by the second derivative test, y is minimum at x = `(1)/(3)` and minimum value of y at x = `(1)/(3)`
= `5(1/3)^3 + 2(1/3)^2 - 3(1/3)`
= `(5)/(27) + (2)/(9) - 1`
= `-(16)/(27)`
Hence, the function has maximum value `(36)/(25)` at x = `-(3)/(5)` and minimum value `-(16)/(27) "at" x = (1)/(3)`.
Solution 2
y = 5x3 + 2x2 – 3x
∴ `dy/dx = d/dx(5x^3 + 2x^2 - 3x)`
= 5 × 3x2 + 2 × 2x – 3 × 1
= 15x2 + 4x – 3
`dy/dx` = 0 gives 15x2 + 4x – 3 = 0
∴ 15x2 + 9x – 5x – 3 = 0
∴ 3x(5x + 3) – 1(5x + 3) = 0
∴ (5x + 3)(3x – 1) = 0
∴ x = `-(3)/(5) or x = (1)/(3)`
∴ the roots of `dy/dx = 0` are `x_1 = -(3)/(5) and x_2 = (1)/(3)`.
First Derivative Test:
(a) `dy/dx` = 15x2 + 4x – 3 = (5x + 3)(3x – 1)
Consider x = `-(3)/(5)`
Let h be a small positive number. Then
`(dy/dx)_("at" x = -3/5 - h) = [5(-3/5 - h) + 3][3 (-3/5 - h) - 1]`
= `( - 3 - 5h + 3)(-9/5 - 3h - 1)`
= `-5h (-14/5 - 3h)`
= `5h (14/5 + 3h) > 0`
and `(dy/dx)_("at" x = -3/5 + h) = [5(-3/5 + h) + 3][3 (-3/5 + h) - 1]`
= `(- 3 + 5h + 3)(-9/5 + 3h - 1)`
= `5h(3h - 14/5) < 0`,
as h is small positive number.
∴ by the first derivative test, y is maximum at x = `-(3)/(5)` and maximum value of y at x = `-(3)/(5)`
= `5(-3/5)^3 + 2(-3/5)^2 - 3(-3/5)`
= `-(27)/(25) + (18)/(25) + (9)/(5) = (36)/(25)`
(b) `dy/dx` = 15x2 + 4x – 3 = (5x + 3)(3x – 1)
Consider x = `(1)/(3)`
Let h be a small positive number. Then
`(dy/dx)_("at" x =1/3 - h) = [5(1/3 - h) + 3][3 (1/3 - h) - 1]`
= `(5/3 - 5h + 3)(1 - 3h - 1)`
= `(14/5 - 5h)(-3h)` < 0, as h is small positive number
and `(dy/dx)_("at" x = 1/3 + h) = [5(1/3 + h) + 3][3 (1/3 + h) - 1]`
= `(5/3 + 5h + 3)(1 + 3h - 1)`
= `(14/3 + 5h)(3h) > 0`
∴ by the first derivative test, y is minimum at x = `(1)/(3)` and minimum value of y at x = `(1)/(3)`
= `5(1/3)^3 + 2(1/3)^2 - 3(1/3)`
= `(5)/(27) + (2)/(9) - 1`
= `(-16)/(27)`
Hence, the function has maximum value `(36)/(25) "at" x = -(3)/(5)` and minimum value `-(16)/(27) "at" x = (1)/(3)`.
APPEARS IN
RELATED QUESTIONS
If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f(x).
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 g(x) = x3 + 1.
Find the maximum and minimum value, if any, of the function given by f(x) = |x + 2| − 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
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`
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) = sin x + cos x , x ∈ [0, π]
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?
It is given that at x = 1, the function x4− 62x2 + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a.
Find the maximum and minimum values of x + sin 2x on [0, 2π].
Show that the right circular cylinder of given surface and maximum volume is such that is heights is equal to the diameter of the base.
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).`
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 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 volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.
Choose the correct option from the given alternatives :
If f(x) = `(x^2 - 1)/(x^2 + 1)`, for every real x, then the minimum value of f is ______.
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.
Solve the following : Show that the height of a right circular cylinder of greatest volume that can be inscribed in a right circular cone is one-third of that of the cone.
Solve the following:
Find the maximum and minimum values of the function f(x) = cos2x + sinx.
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) = ______.
The minimum value of Z = 5x + 8y subject to x + y ≥ 5, 0 ≤ x ≤ 4, y ≥ 2, x ≥ 0, y ≥ 0 is ____________.
If f(x) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.
The minimum value of the function f(x) = 13 - 14x + 9x2 is ______
Show that the function f(x) = 4x3 – 18x2 + 27x – 7 has neither maxima nor minima.
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?
An open box with square base is to be made of a given quantity of cardboard of area c2. Show that the maximum volume of the box is `"c"^3/(6sqrt(3))` cubic units
Find all the points of local maxima and local minima of the function f(x) = (x - 1)3 (x + 1)2
The function f(x) = x5 - 5x4 + 5x3 - 1 has ____________.
Range of projectile will be maximum when angle of projectile is
For all real values of `x`, the minimum value of `(1 - x + x^2)/(1 + x + x^2)`
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.
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 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 ______.
The minimum value of α for which the equation `4/sinx + 1/(1 - sinx)` = α has at least one solution in `(0, π/2)` is ______.
If the function y = `(ax + b)/((x - 4)(x - 1))` has an extremum at P(2, –1), then the values of a and b are ______.
Let f(x) = (x – a)ng(x) , where g(n)(a) ≠ 0; n = 0, 1, 2, 3.... then ______.
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 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 ______.
If f(x) = `1/(4x^2 + 2x + 1); x ∈ R`, then find the maximum value of f(x).
If Mr. Rane order x chairs at the price p = (2x2 - 12x - 192) per chair. How many chairs should he order so that the cost of deal is minimum?
Solution: Let Mr. Rane order x chairs.
Then the total price of x chairs = p·x = (2x2 - 12x- 192)x
= 2x3 - 12x2 - 192x
Let f(x) = 2x3 - 12x2 - 192x
∴ f'(x) = `square` and f''(x) = `square`
f'(x ) = 0 gives x = `square` and f''(8) = `square` > 0
∴ f is minimum when x = 8
Hence, Mr. Rane should order 8 chairs for minimum cost of deal.
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.
Find the point on the curve y2 = 4x, which is nearest to the point (2, 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
