Advertisements
Advertisements
प्रश्न
Discuss the continuity and differentiability of the
Advertisements
उत्तर
\[\text { Given: } f\left( x \right) = \left| x \right| + \left| x - 1 \right|\]
\[\left| x \right| = - x \text { for } x < 0\]
\[\left| x \right| = x \text { for }x > 0\]
\[\left| x - 1 \right| = - \left( x - 1 \right) = - x + 1 \text { for } x - 1 < 0 \text { or }x < 1\]
\[\left| x - 1 \right| = x - 1 \text { for } x - 1 > 0 \text { or }x > 1\]
Now,
\[f\left( x \right) = - x - x + 1 = - 2x + 1 x \in \left( - 1, 0 \right)\]
or
\[f\left( x \right) = x - x + 1 = 1 x \in \left( 0, 1 \right)\]
or
\[f\left( x \right) = x + x - 1 = 2x - 1 x \in \left( 1, 2 \right)\]
Now,
\[\text { LHL } = \lim_{x \to 0^-} f\left( x \right) = \lim_{x \to 0^-} - 2x + 1 = 0 + 1 = 1\]
\[\text { RHL } = \lim_{x \to 0^+} f\left( x \right) = \lim_{x \to 0^+} 1 = 1\]
\[\text { Hence, at x = 0, LHL = RHL}\]
Again,
\[\text { LHL } = \lim_{x \to 1^-} f\left( x \right) = \lim_{x \to 1^-} 1 = 1\]
\[\text { RHL} = \lim_{x \to 1^+} f\left( x \right) = \lim_{x \to 1^+} 2x - 1 = 2 - 1 = 1\]
\[\text { Hence, at x = 1, LHL = RHL}\]
Now,
\[f\left( x \right) = - x - x + 1 = - 2x + 1 x \in \left( - 1, 0 \right)\]
\[ \Rightarrow f'\left( x \right) = - 2 x \in \left( - 1, 0 \right)\]
\[or\]
\[f\left( x \right) = x - x + 1 = 1 x \in \left( 0, 1 \right)\]
\[ \Rightarrow f'\left( x \right) = 0 x \in \left( 0, 1 \right)\]
\[or\]
\[f\left( x \right) = x + x - 1 = 2x - 1 x \in \left( 1, 2 \right)\]
\[ \Rightarrow f'\left( x \right) = 2 x \in \left( 1, 2 \right)\]
Now,
\[\text { LHL }= \lim_{x \to 0^-} f'\left( x \right) = \lim_{x \to 0^-} - 2 = - 2\]
\[\text { RHL }= \lim_{x \to 0^+} f'\left( x \right) = \lim_{x \to 0^+} 0 = 0\]
\[\text { Since, at x = 0, LHL } \neq \text { RHL}\]
\[\text { Hence,} f\left( x \right) \text { is not differentiable at } x = 0\]
Again,
\[\text { LHL }= \lim_{x \to 1^-} f'\left( x \right) = \lim_{x \to 1^-} 0 = 0\]
\[\text { RHL } = \lim_{x \to 1^+} f'\left( x \right) = \lim_{x \to 1^+} 2 = 2\]
\[\text { Since, at } x = 1, \text { LHL } \neq \text { RHL }\]
\[\text { Hence }, f\left( x \right) \text { is not differentiable at } x = 1\]
APPEARS IN
संबंधित प्रश्न
Differentiate the function with respect to x.
sin (ax + b)
Differentiate the function with respect to x.
`sec(tan (sqrtx))`
Differentiate the function with respect to x.
`(sin (ax + b))/cos (cx + d)`
Differentiate the function with respect to x:
(3x2 – 9x + 5)9
Differentiate the function with respect to x:
`(5x)^(3cos 2x)`
Let f(x) = x|x|, for all x ∈ R. Discuss the derivability of f(x) at x = 0
If y = tan(x + y), find `("d"y)/("d"x)`
If y = tanx + secx, prove that `("d"^2y)/("d"x^2) = cosx/(1 - sinx)^2`
Differential coefficient of sec (tan–1x) w.r.t. x is ______.
| COLUMN-I | COLUMN-II |
| (A) If a function f(x) = `{((sin3x)/x, "if" x = 0),("k"/2",", "if" x = 0):}` is continuous at x = 0, then k is equal to |
(a) |x| |
| (B) Every continuous function is differentiable | (b) True |
| (C) An example of a function which is continuous everywhere but not differentiable at exactly one point |
(c) 6 |
| (D) The identity function i.e. f (x) = x ∀ ∈x R is a continuous function |
(d) False |
cos |x| is differentiable everywhere.
Show that the function f(x) = |sin x + cos x| is continuous at x = π.
sinn (ax2 + bx + c)
`cos(tan sqrt(x + 1))`
`sin^-1 1/sqrt(x + 1)`
(sin x)cosx
sinmx . cosnx
(x + 1)2(x + 2)3(x + 3)4
`cos^-1 ((sinx + cosx)/sqrt(2)), (-pi)/4 < x < pi/4`
`tan^-1 (sqrt((1 - cosx)/(1 + cosx))), - pi/4 < x < pi/4`
`tan^-1 ((sqrt(1 + x^2) + sqrt(1 - x^2))/(sqrt(1 + x^2) - sqrt(1 - x^2))), -1 < x < 1, x ≠ 0`
If xm . yn = (x + y)m+n, prove that `("d"^2"y")/("dx"^2)` = 0
If y = `sqrt(sinx + y)`, then `"dy"/"dx"` is equal to ______.
For the curve `sqrt(x) + sqrt(y)` = 1, `"dy"/"dx"` at `(1/4, 1/4)` is ______.
If k be an integer, then `lim_("x" -> "k") ("x" - ["x"])` ____________.
The differential coefficient of `"tan"^-1 ((sqrt(1 + "x") - sqrt (1 - "x"))/(sqrt (1+ "x") + sqrt (1 - "x")))` is ____________.
If `"f"("x") = ("sin" ("e"^("x"-2) - 1))/("log" ("x" - 1)), "x" ne 2 and "f" ("x") = "k"` for x = 2, then value of k for which f is continuous is ____________.
If sin y = x sin (a + y), then value of dy/dx is
A particle is moving on a line, where its position S in meters is a function of time t in seconds given by S = t3 + at2 + bt + c where a, b, c are constant. It is known that at t = 1 seconds, the position of the particle is given by S = 7 m. Velocity is 7 m/s and acceleration is 12 m/s2. The values of a, b, c are ______.
If f(x) = `{{:(x^2"," if x ≥ 1),(x"," if x < 1):}`, then show that f is not differentiable at x = 1.
The set of all points where the function f(x) = x + |x| is differentiable, is ______.
