Advertisements
Advertisements
प्रश्न
f(x) = (x \[-\] 1) (x+2)2.
Advertisements
उत्तर
\[\text { Given }: f\left( x \right) = \left( x - 1 \right) \left( x + 2 \right)^2 \]
\[ \Rightarrow f'\left( x \right) = \left( x + 2 \right)^2 + 2\left( x + 2 \right)\left( x - 1 \right)\]
\[\text{ For a local maximum or a local minimum, we must have }\]
\[f'\left( x \right) = 0\]
\[ \Rightarrow \left( x + 2 \right)^2 + 2\left( x + 2 \right)\left( x - 1 \right) = 0\]
\[ \Rightarrow \left( x + 2 \right)\left( x + 2 + 2x - 2 \right) = 0\]
\[ \Rightarrow \left( x + 2 \right)\left( 3x \right) = 0\]
\[ \Rightarrow x = 0, - 2\]
Since f '(x) changes from negative to positive when x increases through 0, x = 0 is the point of local minima.
The local minimum value of f (x) at x = 0 is given by \[\left( 0 - 1 \right) \left( 0 + 2 \right)^2 = - 4\]
Since f '(x) changes sign from positive to negative when x increases through \[- 2\] ,x = \[- 2\] is the point of local maxima.
The local maximum value of f (x) at x = \[- 2\] is given by
APPEARS IN
संबंधित प्रश्न
f(x) = 4x2 + 4 on R .
f(x) = 16x2 \[-\] 16x + 28 on R ?
f(x) = xex.
`f(x)=xsqrt(32-x^2), -5<=x<=5` .
f(x) = (x \[-\] 1) (x \[-\] 2)2.
`f(x)=xsqrt(1-x), x<=1` .
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 = \[\frac{\pi}{6}\] ?
Find the maximum value of 2x3\[-\] 24x + 107 in the interval [1,3]. Find the maximum value of the same function in [ \[-\] 3, \[-\] 1].
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.
Of all the closed cylindrical cans (right circular), which enclose a given volume of 100 cm3, which has the minimum surface area?
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.
Prove that a conical tent of given capacity will require the least amount of canavas when the height is \[\sqrt{2}\] times the radius of the base.
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 \[\theta\] = \[\frac{\pi}{6}\] .
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is \[6\sqrt{3}\]r.
Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible when revolved about one of its sides ?
A closed cylinder has volume 2156 cm3. What will be the radius of its base so that its total surface area is minimum ?
Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius \[5\sqrt{3 cm} \text { is }500 \pi {cm}^3 .\]
Find the point on the parabolas x2 = 2y which is closest to the point (0,5) ?
Find the coordinates of a point on the parabola y=x2+7x + 2 which is closest to the strainght line y = 3x \[-\] 3 ?
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 ?
A straight line is drawn through a given point P(1,4). Determine the least value of the sum of the intercepts on the coordinate axes ?
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 sufficient conditions for a point x = c to be a point of local maximum.
Write the minimum value of f(x) = xx .
Write the maximum value of f(x) = x1/x.
Let f(x) = x3+3x2 \[-\] 9x+2. Then, f(x) has _________________ .
The minimum value of f(x) = \[x4 - x2 - 2x + 6\] is _____________ .
Let f(x) = (x \[-\] a)2 + (x \[-\] b)2 + (x \[-\] c)2. Then, f(x) has a minimum at x = _____________ .
The point on the curve y2 = 4x which is nearest to, the point (2,1) is _______________ .
If x+y=8, then the maximum value of xy is ____________ .
f(x) = \[\sin + \sqrt{3} \cos x\] is maximum when x = ___________ .
The minimum value of \[\left( x^2 + \frac{250}{x} \right)\] is __________ .
The minimum value of the function `f(x)=2x^3-21x^2+36x-20` is ______________ .
