Advertisements
Advertisements
प्रश्न
f(x) = x3\[-\] 6x2 + 9x + 15
Advertisements
उत्तर
\[\text { Given }: f\left( x \right) = x^3 - 6 x^2 + 9x + 15\]
\[ \Rightarrow f'\left( x \right) = 3 x^2 - 12x + 9\]
\[\text { For the local maxima or minima, we must have }\]
\[ f'\left( x \right) = 0\]
\[ \Rightarrow 3 x^2 - 12x + 9 = 0\]
\[ \Rightarrow x^2 - 4x + 3 = 0\]
\[ \Rightarrow \left( x - 1 \right)\left( x - 3 \right) = 0\]
\[ \Rightarrow x = 1 \text { and } 3\]
\[\text { Thus, x = 1 and x = 3 are the possible points of local maxima or local minima } . \]
\[\text { Now,} \]
\[f''\left( x \right) = 6x - 12\]
\[\text { At }x = 1: \]
\[ f''\left( 1 \right) = 6\left( 1 \right) - 12 = - 6 < 0\]
\[\text {So, x = 1 is the point of local maximum } . \]
\[\text { The local maximum value is given by }\]
\[f\left( 1 \right) = 1^3 - 6 \left( 1 \right)^2 + 9 \times 1 + 15 = 19\]
\[\text { At }x = 3: \]
\[ f''\left( 3 \right) = 6\left( 3 \right) - 12 = 6 > 0\]
\[\text { So, x = 3 is the point of local minimum }. \]
\[\text { The local minimum value is given by }\]
\[f\left( 3 \right) = 3^3 - 6 \left( 3 \right)^2 + 9 \times 3 + 15 = 15\]
APPEARS IN
संबंधित प्रश्न
f(x) = x3 \[-\] 3x.
f(x) = sin 2x, 0 < x < \[\pi\] .
`f(x)=sin2x-x, -pi/2<=x<=pi/2`
`f(x)=2sinx-x, -pi/2<=x<=pi/2`
Find the point of local maximum or local minimum, if any, of the following function, using the first derivative test. Also, find the local maximum or local minimum value, as the case may be:
f(x) = x3(2x \[-\] 1)3.
f(x) =\[\frac{x}{2} + \frac{2}{x} , x > 0\] .
`f(x) = 2/x - 2/x^2, x>0`
f(x) = xex.
`f(x) = x/2+2/x, x>0 `.
`f(x)=xsqrt(32-x^2), -5<=x<=5` .
f(x) = \[x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}\] .
`f(x)=xsqrt(1-x), x<=1` .
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 ?
f(x) = (x \[-\] 1)2 + 3 in [ \[-\] 3,1] ?
`f(x) = 3x^4 - 8x^3 + 12x^2- 48x + 25 " in "[0,3]` .
Find the maximum value of 2x3\[-\] 24x + 107 in the interval [1,3]. Find the maximum value of the same function in [ \[-\] 3, \[-\] 1].
Determine two positive numbers whose sum is 15 and the sum of whose squares is maximum.
A beam is supported at the two end and is uniformly loaded. The bending moment M at a distance x from one end is given by \[M = \frac{Wx}{3}x - \frac{W}{3}\frac{x^3}{L^2}\] .
Find the point at which M is maximum in a given case.
A square piece of tin of side 18 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Find this maximum volume.
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 ?
Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm ?
Find the point on the curve y2 = 4x which is nearest to the point (2,\[-\] 8).
Find the point on the parabolas x2 = 2y which is closest to the point (0,5) ?
Find the maximum slope of the curve y = \[- x^3 + 3 x^2 + 2x - 27 .\]
The total cost of producing x radio sets per day is Rs \[\left( \frac{x^2}{4} + 35x + 25 \right)\] and the price per set at which they may be sold is Rs. \[\left( 50 - \frac{x}{2} \right) .\] Find the daily output to maximum the total profit.
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 total area of a page is 150 cm2. The combined width of the margin at the top and bottom is 3 cm and the side 2 cm. What must be the dimensions of the page in order that the area of the printed matter may be maximum?
Write necessary condition for a point x = c to be an extreme point of the function f(x).
Find the least value of f(x) = \[ax + \frac{b}{x}\], where a > 0, b > 0 and x > 0 .
Write the maximum value of f(x) = x1/x.
For the function f(x) = \[x + \frac{1}{x}\]
Let f(x) = x3+3x2 \[-\] 9x+2. Then, f(x) has _________________ .
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 least value of the function f(x) = \[x3 - 18x2 + 96x\] in the interval [0,9] is _____________ .
The minimum value of \[\left( x^2 + \frac{250}{x} \right)\] is __________ .
If(x) = \[\frac{1}{4x^2 + 2x + 1}\] then its maximum value is _________________ .
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?
