Advertisements
Advertisements
Question
f(x) = (x \[-\] 2) \[\sqrt{x - 1} \text { in }[1, 9]\] .
Advertisements
Solution
\[\text { Given }: f\left( x \right) = \left( x - 2 \right)\sqrt{x - 1}\]
\[ \Rightarrow f'\left( x \right) = \sqrt{x - 1} + \frac{\left( x - 2 \right)}{2\sqrt{x - 1}}\]
\[\text { For a local maximum or a local minimum, we must have }\]
\[f'\left( x \right) = 0\]
\[ \Rightarrow \sqrt{x - 1} + \frac{\left( x - 2 \right)}{2\sqrt{x - 1}} = 0\]
\[ \Rightarrow 2\left( x - 1 \right) + \left( x - 2 \right) = 0\]
\[ \Rightarrow 2x - 2 + x - 2 = 0\]
\[ \Rightarrow 3x - 4 = 0\]
\[ \Rightarrow 3x = 4 \]
\[ \Rightarrow x = \frac{4}{3} \]
\[\text { Thus, the critical points of f are } 1, \frac{4}{3} \text { and } 9 . \]
\[\text { Now }, \]
\[ f\left( 1 \right) = \left( 1 - 2 \right)\sqrt{1 - 1} = 0\]
\[ f\left( \frac{4}{3} \right) = \left( \frac{4}{3} - 2 \right)\sqrt{\frac{4}{3} - 1} = \frac{- 2}{3} \times \frac{1}{\sqrt{3}} = - \frac{2}{3\sqrt{3}}\]
\[ f\left( 9 \right) = \left( 9 - 2 \right)\sqrt{9 - 1} = 14\sqrt{2}\]
\[\text { Hence, the absolute maximum value when x } = 9 \text{ is }14\sqrt{2}\text { and the absolute minimum value when x } = \frac{4}{3} \text{ is }- \frac{2}{3\sqrt{3}} . \]
Notes
The solution given in the book is incorrect . The solution here is created according to the question given in the book
APPEARS IN
RELATED QUESTIONS
f(x)=2x3 +5 on R .
f(x) = x3 (x \[-\] 1)2 .
f(x) = (x \[-\] 1) (x+2)2.
f(x) = \[\frac{1}{x^2 + 2}\] .
f(x) =\[x\sqrt{1 - x} , x > 0\].
f(x) = x4 \[-\] 62x2 + 120x + 9.
f(x) = (x - 1) (x + 2)2.
`f(x) = 2/x - 2/x^2, x>0`
`f(x) = x/2+2/x, x>0 `.
`f(x)=xsqrt(32-x^2), -5<=x<=5` .
f(x) = \[- (x - 1 )^3 (x + 1 )^2\] .
Show that \[\frac{\log x}{x}\] has a maximum value at x = e ?
Find the maximum and minimum values of the function f(x) = \[\frac{4}{x + 2} + x .\]
If f(x) = x3 + ax2 + bx + c has a maximum at x = \[-\] 1 and minimum at x = 3. Determine a, b and c ?
Prove that f(x) = sinx + \[\sqrt{3}\] cosx has maximum value at x = \[\frac{\pi}{6}\] ?
`f(x) = 3x^4 - 8x^3 + 12x^2- 48x + 25 " in "[0,3]` .
Find the absolute maximum and minimum values of the function of given by \[f(x) = \cos^2 x + \sin x, x \in [0, \pi]\] .
How should we choose two numbers, each greater than or equal to `-2, `whose sum______________ so that the sum of the first and the cube of the second is minimum?
Divide 15 into two parts such that the square of one multiplied with the cube of the other is minimum.
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 lengths of the two pieces so that the combined area of the circle and the square is minimum?
A tank with rectangular base and rectangular sides, open at the top, is to the constructed so that its depth is 2 m and volume is 8 m3. If building of tank cost 70 per square metre for the base and Rs 45 per square metre for sides, what is the cost of least expensive tank?
A window in the form of a rectangle is surmounted by a semi-circular opening. The total perimeter of the window is 10 m. Find the dimension of the rectangular of the window to admit maximum light through the whole opening.
Show that among all positive numbers x and y with x2 + y2 =r2, the sum x+y is largest when x=y=r \[\sqrt{2}\] .
Determine the points on the curve x2 = 4y which are nearest to the point (0,5) ?
Find the point on the curve x2 = 8y which is nearest to the point (2, 4) ?
Find the point on the curvey y2 = 2x which is at a minimum distance from the point (1, 4).
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 the maximum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
The maximum value of x1/x, x > 0 is __________ .
For the function f(x) = \[x + \frac{1}{x}\]
The sum of two non-zero numbers is 8, the minimum value of the sum of the reciprocals is ______________ .
At x= \[\frac{5\pi}{6}\] f(x) = 2 sin 3x + 3 cos 3x is ______________ .
The maximum value of f(x) = \[\frac{x}{4 - x + x^2}\] on [ \[-\] 1, 1] is _______________ .
If(x) = x+\[\frac{1}{x}\],x > 0, then its greatest value is _______________ .
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.
Of all the closed right circular cylindrical cans of volume 128π cm3, find the dimensions of the can which has minimum surface area.
Which of the following graph represents the extreme value:-
