Advertisements
Advertisements
Question
f(x) = 16x2 \[-\] 16x + 28 on R ?
Solution
Given: f(x) = 16x2 − 16x + 28
\[\Rightarrow\] f(x) = 4(4x2 - 4x + 1) + 24
\[\Rightarrow\] f(x) = 4(2x − 1)2 + 24
Now,
4(2x − 1)2 \[\geq\] 0 for all x \[\in\] R
\[\Rightarrow\] f(x) = 4(2x − 1)2 + 24 \[\geq\] 24 for all x \[\in\] R
\[\Rightarrow\] f(x)\[\geq\] 24 for all x \[\in\] R.
The minimum value of f is attained when (2x − 1) = 0.
(2x − 1) = 0
⇒ x = \[\frac{1}{2}\]
Therefore, the minimum value of f at x =\[\frac{1}{2}\] is 24.
Since f(x) can be enlarged, the maximum value does not exist, which is evident in the graph also.
Hence, the function f does not have a maximum value.
APPEARS IN
RELATED QUESTIONS
f(x)=| x+2 | on R .
f(x)=sin 2x+5 on R .
f(x)=2x3 +5 on R .
f(x) = \[\frac{1}{x^2 + 2}\] .
f(x) = x3 \[-\] 6x2 + 9x + 15 .
f(x) =\[x\sqrt{1 - x} , x > 0\].
f(x) = xex.
`f(x) = (x+1) (x+2)^(1/3), x>=-2` .
`f(x)=xsqrt(32-x^2), -5<=x<=5` .
f(x) = \[x + \sqrt{1 - x}, x \leq 1\] .
f(x) = \[- (x - 1 )^3 (x + 1 )^2\] .
Find the maximum and minimum values of y = tan \[x - 2x\] .
f(x) = (x \[-\] 1)2 + 3 in [ \[-\] 3,1] ?
Find the absolute maximum and minimum values of a function f given by `f(x) = 12 x^(4/3) - 6 x^(1/3) , x in [ - 1, 1]` .
Find the absolute maximum and minimum values of a function f given by \[f(x) = 2 x^3 - 15 x^2 + 36x + 1 \text { on the interval } [1, 5]\] ?
Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.
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?
Two sides of a triangle have lengths 'a' and 'b' and the angle between them is \[\theta\]. What value of \[\theta\] will maximize the area of the triangle? Find the maximum area of the triangle also.
A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, in cutting off squares from each corners and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum possible?
An isosceles triangle of vertical angle 2 \[\theta\] is inscribed in a circle of radius a. Show that the area of the triangle is maximum when \[\theta\] = \[\frac{\pi}{6}\] .
Find the point on the curvey y2 = 2x which is at a minimum distance from the point (1, 4).
Find the maximum slope of the curve y = \[- x^3 + 3 x^2 + 2x - 27 .\]
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.
A particle is moving in a straight line such that its distance at any time t is given by S = \[\frac{t^4}{4} - 2 t^3 + 4 t^2 - 7 .\] Find when its velocity is maximum and acceleration minimum.
Write necessary condition for a point x = c to be an extreme point of the function f(x).
Write the maximum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
Write the minimum value of f(x) = xx .
Write the maximum value of f(x) = \[\frac{\log x}{x}\], if it exists .
At x= \[\frac{5\pi}{6}\] f(x) = 2 sin 3x + 3 cos 3x is ______________ .
If x lies in the interval [0,1], then the least value of x2 + x + 1 is _______________ .
The least value of the function f(x) = \[x3 - 18x2 + 96x\] in the interval [0,9] is _____________ .
If 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 function f(x) = \[2 x^3 - 15 x^2 + 36x + 4\] is maximum at x = ________________ .
Let f(x) = 2x3\[-\] 3x2\[-\] 12x + 5 on [ 2, 4]. The relative maximum occurs at x = ______________ .
A wire of length 34 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a rectangle whose length is twice its breadth. What should be the lengths of the two pieces, so that the combined area of the square and the rectangle is minimum?
The minimum value of the function `f(x)=2x^3-21x^2+36x-20` is ______________ .
