Advertisements
Advertisements
Question
Let f(x) = x|x|, for all x ∈ R. Discuss the derivability of f(x) at x = 0
Advertisements
Solution
We may rewrite f as f(x) = `{{:(x^2",", "if" x ≥ 0),(-x^2",", "if" x < 0):}`
Now Lf ′(0) = `lim_("h" -> 0^-) ("f"(0 + "h") - "f"(0))/"h"`
= `lim_("h" -> 0^-) (-"h"^2 - 0)/"h"`
= `lim_("h" -> 0^-) - "h"`
= 0
Now Rf ′(0) = `lim_("h" -> 0^+) ("f"(0 + "h") - "f"(0))/"h"`
= `lim_("h" -> 0^+) ("h"^2 - 0)/"h"`
= `lim_("h" -> 0^+) "h"`
= 0
Since the left hand derivative and right hand derivative both are equal, hence f is differentiable at x = 0.
APPEARS IN
RELATED QUESTIONS
Differentiate the function with respect to x.
cos x3 . sin2 (x5)
Differentiate the function with respect to x.
`cos (sqrtx)`
Prove that the function f given by f(x) = |x − 1|, x ∈ R is not differentiable at x = 1.
Differentiate the function with respect to x:
(3x2 – 9x + 5)9
Differentiate the function with respect to x:
`sin^(–1)(xsqrtx), 0 ≤ x ≤ 1`
Differentiate the function with respect to x:
`(cos^(-1) x/2)/sqrt(2x+7)`, −2 < x < 2
If f(x) = |x|3, show that f"(x) exists for all real x and find it.
If y = `[(f(x), g(x), h(x)),(l, m,n),(a,b,c)]`, prove that `dy/dx = |(f'(x), g'(x), h'(x)),(l,m, n),(a,b,c)|`.
If sin y = xsin(a + y) prove that `(dy)/(dx) = sin^2(a + y)/sin a`
If y = tanx + secx, prove that `("d"^2y)/("d"x^2) = cosx/(1 - sinx)^2`
Let f(x)= |cosx|. Then, ______.
If u = `sin^-1 ((2x)/(1 + x^2))` and v = `tan^-1 ((2x)/(1 - x^2))`, then `"du"/"dv"` 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 |
`sin sqrt(x) + cos^2 sqrt(x)`
(x + 1)2(x + 2)3(x + 3)4
`tan^-1 (sqrt((1 - cosx)/(1 + cosx))), - pi/4 < x < pi/4`
`tan^-1 (secx + tanx), - pi/2 < x < pi/2`
If y = `sqrt(sinx + y)`, then `"dy"/"dx"` is equal to ______.
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 `y = (x + sqrt(1 + x^2))^n`, then `(1 + x^2) (d^2y)/(dx^2) + x (dy)/(dx)` is
The rate of increase of bacteria in a certain culture is proportional to the number present. If it doubles in 5 hours then in 25 hours, its number would be
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) = `{{:((sin(p + 1)x + sinx)/x,",", x < 0),(q,",", x = 0),((sqrt(x + x^2) - sqrt(x))/(x^(3//2)),",", x > 0):}`
is continuous at x = 0, then the ordered pair (p, q) is equal to ______.
Let S = {t ∈ R : f(x) = |x – π| (e|x| – 1)sin |x| is not differentiable at t}. Then the set S is equal to ______.
If f(x) = `{{:(ax + b; 0 < x ≤ 1),(2x^2 - x; 1 < x < 2):}` is a differentiable function in (0, 2), then find the values of a and b.
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 ______.
Differentiability determines whether a function has what at a particular point?
If \[u\] and \[v\] are differentiable functions, what is \[(uv)'\]?
For \[v\ne0\], what is the derivative of \[\frac{u}{v}\]?
What is \[\frac{d}{dx}(x^n)\]?
What is the practical condition for differentiability at a point?
For differentiability on a closed interval \[[a,b]\], which derivative is considered at \[a\]?
If \[f\] is differentiable at \[c\], which limit equals \[f'(c)\]?
For \[x\ne c\], which identity is used to prove that differentiability implies continuity?
For \[f(x)=|x|\], what is the right-hand derivative at \[x=0\]?
