Find the intervals in which the function f given by \(f(x)=x^3+\frac{1}{x^3},x≠0\) is (i)increasing (ii)decreasing
Solution and Explanation
\(f(x)=x^3+\frac{1}{x^3}\)
∴\(f(x)=3x^2-\frac{3}{x^4}=\frac{3x^6-3}{x^4}\)
Then, \(f'(x)=0\)
\(⇒3x^6-3=0\)
\(⇒x^6=1\)
\(⇒x=±1\)
Now, the points x=1 and x=−1 divide the real line into three disjoint intervals i.e.,\((-∞,-1),(-1,1)\) and \((1,∞)\).
In intervals \((-∞,-1)\) and \((1,∞)\) i.e., when x<−1 and x>1, \(f'(x)>0\)
Thus, when x<−1 and x>1, f is increasing.
In interval (−1,1) i.e., when \(−1<x<1,f'(x)<0\)
Thus, when −1<x<1, f is decreasing.
Top CBSE CLASS XII Mathematics Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A committee of 11 members is to be formed from 8 males and 5 females. Let m denotes the number of ways of selecting committee having atleast 6 males and n denotes the number of ways of selecting atleast 3 females then
Determine whether each of the following relations are reflexive, symmetric, and transitive.
- Relation R in the set A={1,2,3,...13,14} defined as R={(x,y): 3x-y=0}.
- Relation R in the set of N natural numbers defined as R={(x,y): y=x+5 and x<4}.
- Relation R in the set A={1,2,3,4,5,6} defined as R={(x,y): y is divisible by x}.
- Relation R in the set of Z integers defined as R={(x,y): x-y is an integer}
- Relation R in the set of human beings in a town at a particular time given by
- R={(x,y): x and y work at same place}
- R={(x,y): x and y live in the same locality}
- R={(x,y): x is exactly 7 am taller that y}
- R={(x,y): x is wife of y}
- R={(x,y):x is father of y}
Show that the relation R in the set R of real numbers, defined as
R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.
Top CBSE CLASS XII Applications of Derivatives Questions
- Show that of all the rectangles inscribed in a given fixed circle,the square has the maximum area.
- Find the rate of change of the area of a circle with respect to its radius r when (a) r=3 cm (b) r=4 cm
- The volume of a cube is increasing at the rate of \(8 cm^3 /s\). How fast is the surface area increasing when the length of an edge is \(12 cm\)?
- The radius of a circle is increasing uniformly at the rate of \(3 cm/s\). Find the rate at which the area of the circle is increasing when the radius is \(10 cm\).
- An edge of a variable cube is increasing at the rate of \(3 cm/s\). How fast is the volume of the cube increasing when the edge is \(10 cm\) long?
Top CBSE CLASS XII Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A boy of mass 50 kg is standing at one end of a, boat of length 9 m and mass 400 kg. He runs to the other, end. The distance through which the centre of mass of the boat boy system moves is
- A capillary tube of radius r is dipped inside a large vessel of water. The mass of water raised above water level is M. If the radius of capillary is doubled, the mass of water inside capillary will be
- A circular disc is rotating about its own axis. An external opposing torque 0.02 Nm is applied on the disc by which it comes rest in 5 seconds. The initial angular momentum of disc is
- A circular disc is rotating about its own axis at uniform angular velocity \(\omega.\) The disc is subjected to uniform angular retardation by which its angular velocity is decreased to \(\frac {\omega}{2}\) during 120 rotations. The number of rotations further made by it before coming to rest is