Advertisements
Advertisements
प्रश्न
If f (x) = |3 − x| + (3 + x), where (x) denotes the least integer greater than or equal to x, then f (x) is
विकल्प
continuous and differentiable at x = 3
continuous but not differentiable at x = 3
differentiable nut not continuous at x = 3
neither differentiable nor continuous at x = 3
Advertisements
उत्तर
(d) neither differentiable nor continuous at x = 3
We have,
\[f\left( x \right) = \left| 3 - x \right| + \left( 3 + x \right), \text { where } \left( x \right) \text{denotes the least integer greater than or equal to} x . \]
`f(x) = {(3-x +3+3,2<x<3),(-3 +x + 3 +4,3<x<4):}`
`⇒ f(x) = {(-x +9,2<x<3),(x+4 , 3<x<4):}`
Here,
\[\left( \text { LHL at x } = 3 \right) = \lim_{x \to 3^-} f\left( x \right) = \lim_{x \to 3^-} \left( - x + 9 \right) = - 3 + 9 = 6\]
\[\left( \text { RHL at x }= 3 \right) = \lim_{x \to 3^+} f\left( x \right) = \lim_{x \to 3^-} \left( x + 4 \right) = 3 + 4 = 7\]
\[\text { Since, } \left( \text { LHL at x } = 3 \right) \neq \left( \text { RHL at x }= 3 \right)\]
\[\text{Hence, given function is not continuous at x} = 3\]
\[\text{Therefore, the function will also not be differentiable at} x = 3\]
APPEARS IN
संबंधित प्रश्न
Examine the continuity of the following function :
`{:(,,f(x)= x^2 -x+9,"for",x≤3),(,,=4x+3,"for",x>3):}}"at "x=3`
Examine the following function for continuity:
f(x) = `(x^2 - 25)/(x + 5)`, x ≠ −5
Discuss the continuity of the function f, where f is defined by:
f(x) = `{(-2", if" x <= -1),(2x", if" -1 < x <= 1),(2", if" x > 1):}`
Determine the value of the constant k so that the function
\[f\left( x \right) = \begin{cases}k x^2 , if & x \leq 2 \\ 3 , if & x > 2\end{cases}\text{is continuous at x} = 2 .\]
For what value of k is the function
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point;
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}kx + 1, if & x \leq 5 \\ 3x - 5, if & x > 5\end{cases}\] at x = 5
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}k( x^2 + 2), \text{if} & x \leq 0 \\ 3x + 1 , \text{if} & x > 0\end{cases}\]
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{x^4 + x^3 + 2 x^2}{\tan^{- 1} x}, & \text{ if } x \neq 0 \\ 10 , & \text{ if } x = 0\end{cases}\]
Write the value of b for which \[f\left( x \right) = \begin{cases}5x - 4 & 0 < x \leq 1 \\ 4 x^2 + 3bx & 1 < x < 2\end{cases}\] is continuous at x = 1.
If the function \[f\left( x \right) = \begin{cases}\left( \cos x \right)^{1/x} , & x \neq 0 \\ k , & x = 0\end{cases}\] is continuous at x = 0, then the value of k is
The value of f (0) so that the function
The function \[f\left( x \right) = \frac{x^3 + x^2 - 16x + 20}{x - 2}\] is not defined for x = 2. In order to make f (x) continuous at x = 2, Here f (2) should be defined as ______.
The values of the constants a, b and c for which the function \[f\left( x \right) = \begin{cases}\left( 1 + ax \right)^{1/x} , & x < 0 \\ b , & x = 0 \\ \frac{\left( x + c \right)^{1/3} - 1}{\left( x + 1 \right)^{1/2} - 1}, & x > 0\end{cases}\] may be continuous at x = 0, are
The points of discontinuity of the function\[f\left( x \right) = \begin{cases}\frac{1}{5}\left( 2 x^2 + 3 \right) , & x \leq 1 \\ 6 - 5x , & 1 < x < 3 \\ x - 3 , & x \geq 3\end{cases}\text{ is } \left( are \right)\]
Show that the function f defined as follows, is continuous at x = 2, but not differentiable thereat:
Show that the function
\[f\left( x \right) = \begin{cases}\left| 2x - 3 \right| \left[ x \right], & x \geq 1 \\ \sin \left( \frac{\pi x}{2} \right), & x < 1\end{cases}\] is continuous but not differentiable at x = 1.
Write an example of a function which is everywhere continuous but fails to differentiable exactly at five points.
Define differentiability of a function at a point.
Let f (x) = |x| and g (x) = |x3|, then
Let \[f\left( x \right) = \begin{cases}\frac{1}{\left| x \right|} & for \left| x \right| \geq 1 \\ a x^2 + b & for \left| x \right| < 1\end{cases}\] If f (x) is continuous and differentiable at any point, then
Let f (x) = |sin x|. Then,
Evaluate :`int Sinx/(sqrt(cos^2 x-2 cos x-3)) dx`
If f is continuous at x = 0, then find f (0).
Where f(x) = `(3^"sin x" - 1)^2/("x" . "log" ("x" + 1)) , "x" ≠ 0`
If the function f is continuous at = 2, then find f(2) where f(x) = `(x^5 - 32)/(x - 2)`, for ≠ 2.
If f(x) = `(e^(2x) - 1)/(ax)` . for x < 0 , a ≠ 0
= 1. for x = 0
= `(log(1 + 7x))/(bx)`. for x > 0 , b ≠ 0
is continuous at x = 0 . then find a and b
Examine the continuity of the following function :
`{:(,f(x),=(x^2-16)/(x-4),",","for "x!=4),(,,=8,",","for "x=4):}} " at " x=4`
Examine the differentiability of the function f defined by
f(x) = `{{:(2x + 3",", "if" -3 ≤ x < - 2),(x + 1",", "if" -2 ≤ x < 0),(x + 2",", "if" 0 ≤ x ≤ 1):}`
y = |x – 1| is a continuous function.
f(x) = `{{:((2^(x + 2) - 16)/(4^x - 16)",", "if" x ≠ 2),("k"",", "if" x = 2):}` at x = 2
Find the values of a and b such that the function f defined by
f(x) = `{{:((x - 4)/(|x - 4|) + "a"",", "if" x < 4),("a" + "b"",", "if" x = 4),((x - 4)/(|x - 4|) + "b"",", "if" x > 4):}`
is a continuous function at x = 4.
Given the function f(x) = `1/(x + 2)`. Find the points of discontinuity of the composite function y = f(f(x))
Show that f(x) = |x – 5| is continuous but not differentiable at x = 5.
The set of points where the function f given by f(x) = |2x − 1| sinx is differentiable is ______.
`lim_("x" -> 0) (2 "sin x - sin" 2 "x")/"x"^3` is equal to ____________.
