Advertisements
Advertisements
Question
f(x) = x3\[-\] 6x2 + 9x + 15
Advertisements
Solution
\[\text { Given }: f\left( x \right) = x^3 - 6 x^2 + 9x + 15\]
\[ \Rightarrow f'\left( x \right) = 3 x^2 - 12x + 9\]
\[\text { For the local maxima or minima, we must have }\]
\[ f'\left( x \right) = 0\]
\[ \Rightarrow 3 x^2 - 12x + 9 = 0\]
\[ \Rightarrow x^2 - 4x + 3 = 0\]
\[ \Rightarrow \left( x - 1 \right)\left( x - 3 \right) = 0\]
\[ \Rightarrow x = 1 \text { and } 3\]
\[\text { Thus, x = 1 and x = 3 are the possible points of local maxima or local minima } . \]
\[\text { Now,} \]
\[f''\left( x \right) = 6x - 12\]
\[\text { At }x = 1: \]
\[ f''\left( 1 \right) = 6\left( 1 \right) - 12 = - 6 < 0\]
\[\text {So, x = 1 is the point of local maximum } . \]
\[\text { The local maximum value is given by }\]
\[f\left( 1 \right) = 1^3 - 6 \left( 1 \right)^2 + 9 \times 1 + 15 = 19\]
\[\text { At }x = 3: \]
\[ f''\left( 3 \right) = 6\left( 3 \right) - 12 = 6 > 0\]
\[\text { So, x = 3 is the point of local minimum }. \]
\[\text { The local minimum value is given by }\]
\[f\left( 3 \right) = 3^3 - 6 \left( 3 \right)^2 + 9 \times 3 + 15 = 15\]
APPEARS IN
RELATED QUESTIONS
f(x) = 16x2 \[-\] 16x + 28 on R ?
f(x) = x3 \[-\] 1 on R .
f(x) = x3 \[-\] 3x.
f(x) = (x \[-\] 1) (x+2)2.
f(x) = sin x \[-\] cos x, 0 < x < 2\[\pi\] .
`f(x) = 2/x - 2/x^2, x>0`
`f(x) = x/2+2/x, x>0 `.
`f(x) = (x+1) (x+2)^(1/3), x>=-2` .
f(x) = \[x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}\] .
f(x) = \[x + \sqrt{1 - x}, x \leq 1\] .
`f(x)=xsqrt(1-x), x<=1` .
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) = 4x \[-\] \[\frac{x^2}{2}\] in [ \[-\] 2,4,5] .
f(x) = (x \[-\] 1)2 + 3 in [ \[-\] 3,1] ?
Find the absolute maximum and minimum values of a function f given by f(x) = 2x3 − 15x2 + 36x + 1 on the interval [1, 5].
Divide 64 into two parts such that the sum of the cubes of two parts is minimum.
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?
Of all the closed cylindrical cans (right circular), which enclose a given volume of 100 cm3, which has the minimum surface area?
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?
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.
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is \[6\sqrt{3}\]r.
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 curvey y2 = 2x which is at a minimum distance from the point (1, 4).
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 the minimum value of f(x) = \[x + \frac{1}{x}, x > 0 .\]
The minimum value of f(x) = \[x4 - x2 - 2x + 6\] is _____________ .
The number which exceeds its square by the greatest possible quantity is _________________ .
Let f(x) = (x \[-\] a)2 + (x \[-\] b)2 + (x \[-\] c)2. Then, f(x) has a minimum at x = _____________ .
The function f(x) = \[\sum^5_{r = 1}\] (x \[-\] r)2 assumes minimum value at x = ______________ .
The least value of the function f(x) = \[x3 - 18x2 + 96x\] in the interval [0,9] is _____________ .
The least and greatest values of f(x) = x3\[-\] 6x2+9x in [0,6], are ___________ .
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 ______________ .
Let x, y be two variables and x>0, xy=1, then minimum value of x+y 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?
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.
Which of the following graph represents the extreme value:-
