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) = x3 − 3x
Advertisements
उत्तर
Given function g(x) = x3 - 3x
`therefore g'(x) = 3x^2 - 3`
if, g'(x) = 0 and 3x2 - 3 = 0
⇒ x2 - 1 = 0
⇒ x = `pm` 1
The points at which extremum may occurs are -1 and +1.
g' (x) = 6x
g' (-1) = 6 (-1) = -6 < 0
∴ g has a local maximum at x = -1 and local maxum value at x = -1 is g (-1) = (-1)3 - 3 (-1)
= -1 + 3
= 2
g' (1) = 6 × 1
= 6 > 0
∴g has a local minimum at x = 1 and local minimum value at x = 1 is g (1)
= 13 - 3 × 1
= -2
APPEARS IN
संबंधित प्रश्न
Examine the maxima and minima of the function f(x) = 2x3 - 21x2 + 36x - 20 . Also, find the maximum and minimum values of f(x).
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) = 1/(x^2 + 2)`
Prove that the following function do not have maxima or minima:
f(x) = ex
A square piece of tin of side 18 cm is to made into a box without a top by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum 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 right circular cone of least curved surface and given volume has an altitude equal to `sqrt2` time the radius 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).`
Find the maximum and minimum of the following functions : f(x) = `x^2 + (16)/x^2`
Divide the number 30 into two parts such that their product is maximum.
Divide the number 20 into two parts such that sum of their squares is minimum.
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)`.
Solve the following:
Find the maximum and minimum values of the function f(x) = cos2x + sinx.
A metal wire of 36 cm length is bent to form a rectangle. Find its dimensions when its area is maximum.
The total cost of producing x units is ₹ (x2 + 60x + 50) and the price is ₹ (180 − x) per unit. For what units is the profit maximum?
Find the local maximum and local minimum value of f(x) = x3 − 3x2 − 24x + 5
Maximum slope of the curve y = –x3 + 3x2 + 9x – 27 is ______.
Find all the points of local maxima and local minima of the function f(x) = (x - 1)3 (x + 1)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.
If y = x3 + x2 + x + 1, then y ____________.
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)
The point on the curve `x^2 = 2y` which is nearest to the point (0, 5) is
A function f(x) is maximum at x = a when f'(a) > 0.
If y = alog|x| + bx2 + x has its extremum values at x = –1 and x = 2, then ______.
The function g(x) = `(f(x))/x`, x ≠ 0 has an extreme value when ______.
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 ______.
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.
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 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.
For a function defined on an interval \[I\], which condition means that \[f\] has a minimum value at \[c\in I\]?
Let \[c\] be a critical point of a continuous function \[f\]. If \[f'(x)\] changes sign from positive to negative as \[x\] passes through \[c\], what is \[c\]?
What are the critical points obtained from \[12x(x-1)(x+2)=0\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], what conclusion follows at \[x=0\]?
For \[f(x)=3x^4+4x^3-12x^2+12\], what conclusion follows at \[x=1\]?
