Advertisements
Advertisements
प्रश्न
Show that f(x) = x1/3 is not differentiable at x = 0.
उत्तर
Disclaimer: It might be a wrong question because f(x) is differentiable at x=0
Given:
We have,
(LHD at x = 0)
\[\lim_{x \to 0^-} \frac{f(x) - f(0)}{x - 0}\]
\[ = \lim_{h \to 0} \frac{f(0 - h) - f(0)}{0 - h - 0}\]
\[ = \lim_{h \to 0} \frac{\left( 0 - h \right)^\frac{1}{3} - 0^\frac{1}{3}}{- h}\]
\[ = \lim_{h \to 0} \frac{\left( - h \right)^\frac{1}{3}}{- h}\]
\[ = \lim_{h \to 0} \left( - h \right)^\frac{- 2}{3} \]
\[ = 0\]
(RHD at x = 0)
\[\lim_{x \to 0^+} \frac{f(x) - f(0)}{x - 0}\]
\[ = \lim_{h \to 0} \frac{f(0 + h) - f(0)}{0 + h - 0}\]
\[ = \lim_{h \to 0} \frac{\left( 0 + h \right)^\frac{1}{3} - 0^\frac{1}{3}}{- h}\]
\[ = \lim_{h \to 0} \frac{h^\frac{1}{3}}{h}\]
\[ = \lim_{h \to 0} h^\frac{- 2}{3} \]
\[ = 0\]
LHD at (x = 0)= RHD at (x = 0)
Hence,
APPEARS IN
संबंधित प्रश्न
Discuss the continuity of the function f, where f is defined by `f(x) = {(2x , ","if x < 0),(0, "," if 0 <= x <= 1),(4x, "," if x > 1):}`
A function f(x) is defined as,
Discuss the continuity of the following functions at the indicated point(s):
(ii) \[f\left( x \right) = \left\{ \begin{array}{l}x^2 \sin\left( \frac{1}{x} \right), & x \neq 0 \\ 0 , & x = 0\end{array}at x = 0 \right.\]
Discuss the continuity of the following functions at the indicated point(s):
Show that
\[f\left( x \right) = \begin{cases}1 + x^2 , if & 0 \leq x \leq 1 \\ 2 - x , if & x > 1\end{cases}\]
Determine the value of the constant k so that the function
\[f\left( x \right) = \left\{ \begin{array}{l}\frac{x^2 - 3x + 2}{x - 1}, if & x \neq 1 \\ k , if & x = 1\end{array}\text{is continuous at x} = 1 \right.\]
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \begin{cases}\frac{x^2 - 25}{x - 5}, & x \neq 5 \\ k , & x = 5\end{cases}\]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; \[f\left( x \right) = \begin{cases}k( x^2 + 2), \text{if} & x \leq 0 \\ 3x + 1 , \text{if} & x > 0\end{cases}\]
In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; \[f\left( x \right) = \binom{\frac{x^3 + x^2 - 16x + 20}{\left( x - 2 \right)^2}, x \neq 2}{k, x = 2}\]
Discuss the continuity of the f(x) at the indicated points:
(i) f(x) = | x | + | x − 1 | at x = 0, 1.
Find all the points of discontinuity of f defined by f (x) = | x |− | x + 1 |.
Define continuity of a function at a point.
If \[f\left( x \right) = \left| \log_{10} x \right|\] then at x = 1
If \[f\left( x \right) = \begin{cases}\frac{\sin (a + 1) x + \sin x}{x} , & x < 0 \\ c , & x = 0 \\ \frac{\sqrt{x + b x^2} - \sqrt{x}}{bx\sqrt{x}} , & x > 0\end{cases}\]is continuous at x = 0, then
If \[f\left( x \right) = \left\{ \begin{array}a x^2 + b , & 0 \leq x < 1 \\ 4 , & x = 1 \\ x + 3 , & 1 < x \leq 2\end{array}, \right.\] then the value of (a, b) for which f (x) cannot be continuous at x = 1, is
Show that the function f defined as follows, is continuous at x = 2, but not differentiable thereat:
If f is defined by f (x) = x2, find f'(2).
Define differentiability of a function at a point.
Is every continuous function differentiable?
If f (x) is differentiable at x = c, then write the value of
Let \[f\left( x \right) = \left( x + \left| x \right| \right) \left| x \right|\]
Let f (x) = |x| and g (x) = |x3|, then
If \[f\left( x \right) = \left| \log_e x \right|, \text { then}\]
If f (x) = |3 − x| + (3 + x), where (x) denotes the least integer greater than or equal to x, then f (x) is
If \[f\left( x \right) = \begin{cases}\frac{1}{1 + e^{1/x}} & , x \neq 0 \\ 0 & , x = 0\end{cases}\] then f (x) is
Find k, if f(x) =`log (1+3x)/(5x)` for x ≠ 0
= k for x = 0
is continuous at x = 0.
Find the value of k for which the function f (x ) = \[\binom{\frac{x^2 + 3x - 10}{x - 2}, x \neq 2}{ k , x^2 }\] is continuous at x = 2 .
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.
The probability distribution function of continuous random variable X is given by
f( x ) = `x/4`, 0 < x < 2
= 0, Otherwise
Find P( x ≤ 1)
Discuss the continuity of the function `f(x) = (3 - sqrt(2x + 7))/(x - 1)` for x ≠ 1
= `-1/3` for x = 1, at x = 1
If the function
f(x) = x2 + ax + b, x < 2
= 3x + 2, 2≤ x ≤ 4
= 2ax + 5b, 4 < x
is continuous at x = 2 and x = 4, then find the values of a and b
Show that the function f defined by f(x) = `{{:(x sin 1/x",", x ≠ 0),(0",", x = 0):}` is continuous at x = 0.
Let f(x) = `{{:((1 - cos 4x)/x^2",", "if" x < 0),("a"",", "if" x = 0),(sqrt(x)/(sqrt(16) + sqrt(x) - 4)",", "if" x > 0):}`. For what value of a, f is continuous at x = 0?
The value of k which makes the function defined by f(x) = `{{:(sin 1/x",", "if" x ≠ 0),("k"",", "if" x = 0):}`, continuous at x = 0 is ______.
f(x) = `{{:(3x + 5",", "if" x ≥ 2),(x^2",", "if" x < 2):}` at x = 2
f(x) = `{{:(|x - 4|/(2(x - 4))",", "if" x ≠ 4),(0",", "if" x = 4):}` at x = 4
f(x) = `{{:(x^2/2",", "if" 0 ≤ x ≤ 1),(2x^2 - 3x + 3/2",", "if" 1 < x ≤ 2):}` at x = 1
Given functions `"f"("x") = ("x"^2 - 4)/("x" - 2) "and g"("x") = "x" + 2, "x" le "R"`. Then which of the following is correct?
