Advertisements
Advertisements
प्रश्न
Find the absolute maximum and minimum values of a function f given by f(x) = 2x3 − 15x2 + 36x + 1 on the interval [1, 5].
Find the absolute maximum and absolute minimum of function f(x) = 2x3 − 15x2 + 36x + 1 on [1, 5].
Advertisements
उत्तर
Given: f(x) = 2x3 − 15x2 + 36x + 1
Differentiate w. r. t. x,
f'(x) = 6x2 − 30x + 36
Put f'(x) = 0
6(x2 − 5x + 6) = 0
x2 − 5x + 6 = 0
x2 − 3x − 2x + 6 = 0
x(x − 3) −2(x − 3) = 0
(x − 3)(x − 2) = 0
then, x =3 or 2
x = 1, 2, 3, 5
Now f(1) = 2(1)3 – 15(1)2 + 36(1) + 1
= 2 – 15 + 36 + 1
= 24
f(2) = 2(2)3 – 15(2)2 + 36(2) + 1
= 16 – 60 + 72 + 1
= 29
f(3) = 2(3)3 – 15(3)2 + 36(3) + 1
= 54 – 135 + 108 + 1
= 28
f(5) = 2(5)3 – 15(5)2 + 36(5) + 1
= 250 – 375 + 180 + 1
= 56
Hence, the absolute maximum value when x = 5 is 56, and the absolute minimum value when x = 1 is 24.
APPEARS IN
संबंधित प्रश्न
f(x) = x3 \[-\] 1 on R .
f(x) = (x \[-\] 1) (x+2)2.
f(x) = x3 \[-\] 6x2 + 9x + 15 .
f(x) = cos x, 0 < x < \[\pi\] .
`f(x)=2sinx-x, -pi/2<=x<=pi/2`
f(x) = (x - 1) (x + 2)2.
`f(x) = x/2+2/x, x>0 `.
f(x) = (x \[-\] 1) (x \[-\] 2)2.
f(x) = \[- (x - 1 )^3 (x + 1 )^2\] .
The function y = a log x+bx2 + x has extreme values at x=1 and x=2. Find a and b ?
Prove that f(x) = sinx + \[\sqrt{3}\] cosx has maximum value at x = \[\frac{\pi}{6}\] ?
f(x) = 4x \[-\] \[\frac{x^2}{2}\] in [ \[-\] 2,4,5] .
`f(x) = 3x^4 - 8x^3 + 12x^2- 48x + 25 " in "[0,3]` .
f(x) = (x \[-\] 2) \[\sqrt{x - 1} \text { in }[1, 9]\] .
Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.
Divide 15 into two parts such that the square of one multiplied with the cube of the other is minimum.
Of all the closed cylindrical cans (right circular), which enclose a given volume of 100 cm3, which has the minimum surface area?
A wire of length 20 m is to be cut into two pieces. One of the pieces will be bent into shape of a square and the other into shape of an equilateral triangle. Where the we should be cut so that the sum of the areas of the square and triangle is minimum?
Given the sum of the perimeters of a square and a circle, show that the sum of there areas is least when one side of the square is equal to diameter of the circle.
Find the largest possible area of a right angled triangle whose hypotenuse is 5 cm long.
A rectangle is inscribed in a semi-circle of radius r with one of its sides on diameter of semi-circle. Find the dimension of the rectangle so that its area is maximum. Find also the area ?
An open tank is to be constructed with a square base and vertical sides so as to contain a given quantity of water. Show that the expenses of lining with lead with be least, if depth is made half of width.
The strength of a beam varies as the product of its breadth and square of its depth. Find the dimensions of the strongest beam which can be cut from a circular log of radius a ?
Write sufficient conditions for a point x = c to be a point of local maximum.
Write the maximum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
Find the least value of f(x) = \[ax + \frac{b}{x}\], where a > 0, b > 0 and x > 0 .
If \[ax + \frac{b}{x} \frac{>}{} c\] for all positive x where a,b,>0, then _______________ .
The minimum value of f(x) = \[x4 - x2 - 2x + 6\] is _____________ .
The sum of two non-zero numbers is 8, the minimum value of the sum of the reciprocals is ______________ .
f(x) = \[\sin + \sqrt{3} \cos x\] is maximum when x = ___________ .
The minimum value of \[\left( x^2 + \frac{250}{x} \right)\] is __________ .
Let x, y be two variables and x>0, xy=1, then minimum value of x+y is _______________ .
The maximum value of f(x) = \[\frac{x}{4 + x + x^2}\] on [ \[-\] 1,1] is ___________________ .
Let f(x) = 2x3\[-\] 3x2\[-\] 12x + 5 on [ 2, 4]. The relative maximum occurs at x = ______________ .
The sum of the surface areas of a cuboid with sides x, 2x and \[\frac{x}{3}\] 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 sphere. Also find the minimum value of the sum of their volumes.
