Advertisements
Advertisements
प्रश्न
The function f(x) = x |x| is ______.
विकल्प
continuous and differentiable at x = 0
continuous but not differentiable at x = 0
differentiable but not continuous at x = 0
neither differentiable nor continuous at x = 0
Advertisements
उत्तर
The function f(x) = x |x| is continuous and differentiable at x = 0.
Explanation:
Given, the function is f(x) = x |x| for x ∈ R.
The function can be written as,
f(x) = `{{:( x^2; x > 0),(-x^2; x ≤ 0):}`
Now, Rf(0) = `lim_(x rightarrow 0^+) (x^2)` = 0
and Lf(0) = `lim_(x rightarrow 0^-) (-x^2)` = 0
So, Lf(0) = Rf(0) = f(0)
So, the function is continuous at 0.
Now, Rf'(0) = `lim_(x rightarrow 0^+) (f(x) - f(0))/(x - 0)`
= `lim_(x rightarrow 0^+) (x^2 - 0)/x` = 0
and Lf'(0) = `lim_(x rightarrow 0^-) (f(x) - f(0))/(x - 0)`
= `lim_(x rightarrow 0^-) (-x^2 - 0)/x` = 0
So, Lf'(0) = Rf'(0)
So, function is differentiable at 0.
संबंधित प्रश्न
Find the relationship between a and b so that the function f defined by f(x) = `{(ax + 1", if" x<= 3),(bx + 3", if" x > 3):}` is continuous at x = 3.
Find the value of k so that the function f is continuous at the indicated point.
f(x) = `{((kcosx)/(pi-2x)", if" x != pi/2),(3", if" x = pi/2):}` at x = `"pi/2`
Find the value of k so that the function f is continuous at the indicated point.
f(x) = `{(kx +1", if" x<= pi),(cos x", if" x > pi):}` at x = π
Find the value of k so that the function f is continuous at the indicated point.
f(x) = `{(kx + 1", if" x <= 5),(3x - 5", if" x > 5):}` at x = 5
Find the values of a and b such that the function defined by f(x) = `{(5", if" x <= 2),(ax +b", if" 2 < x < 10),(21", if" x >= 10):}` is a continuous function.
Extend the definition of the following by continuity
Prove that the function \[f\left( x \right) = \begin{cases}\frac{\sin x}{x}, & x < 0 \\ x + 1, & x \geq 0\end{cases}\] is everywhere continuous.
Find the points of discontinuity, if any, of the following functions:
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{x^4 - 16}{x - 2}, & \text{ if } x \neq 2 \\ 16 , & \text{ if } x = 2\end{cases}\]
Find the points of discontinuity, if any, of the following functions: \[f\left( x \right) = \begin{cases}\frac{\sin x}{x} + \cos x, & \text{ if } x \neq 0 \\ 5 , & \text { if } x = 0\end{cases}\]
What happens to a function f (x) at x = a, if
If \[f\left( x \right) = \binom{\frac{1 - \cos x}{x^2}, x \neq 0}{k, x = 0}\] is continuous at x = 0, find k.
Determine the value of the constant 'k' so that function f
Let \[f\left( x \right) = \left\{ \begin{array}\\ \frac{x - 4}{\left| x - 4 \right|} + a, & x < 4 \\ a + b , & x = 4 \\ \frac{x - 4}{\left| x - 4 \right|} + b, & x > 4\end{array} . \right.\]Then, f (x) is continuous at x = 4 when
If the function f (x) defined by \[f\left( x \right) = \begin{cases}\frac{\log \left( 1 + 3x \right) - \log \left( 1 - 2x \right)}{x}, & x \neq 0 \\ k , & x = 0\end{cases}\] is continuous at x = 0, then k =
The value of a for which the function \[f\left( x \right) = \begin{cases}5x - 4 , & \text{ if } 0 < x \leq 1 \\ 4 x^2 + 3ax, & \text{ if } 1 < x < 2\end{cases}\] is continuous at every point of its domain, is
Find the values of a and b, if the function f defined by
If \[f\left( x \right) = \begin{cases}\frac{\left| x + 2 \right|}{\tan^{- 1} \left( x + 2 \right)} & , x \neq - 2 \\ 2 & , x = - 2\end{cases}\] then f (x) is
Let f (x) = a + b |x| + c |x|4, where a, b, and c are real constants. Then, f (x) is differentiable at x = 0, if
`lim_("x" -> 0) ("x cos x" - "log" (1 + "x"))/"x"^2` is equal to ____________.
The value of f(0) for the function `f(x) = 1/x[log(1 + x) - log(1 - x)]` to be continuous at x = 0 should be
The function f(x) = 5x – 3 is continuous at x =
The function f(x) = x2 – sin x + 5 is continuous at x =
The value of ‘k’ for which the function f(x) = `{{:((1 - cos4x)/(8x^2)",", if x ≠ 0),(k",", if x = 0):}` is continuous at x = 0 is ______.
Which conditions guarantee that \[f\circ g\] is continuous at \[c\]?
Why is \[g(x)=1-x+|x|\] continuous?
Why is \[f(x)=|1-x+|x||\] continuous for every real \[x\]?
