If x + | y | = 2y, th e n y as a function of x is
- defined for all real x
- continuous at x = 0
- differentiable for all x
- such that $ \frac{dy}{dx} = \frac{1}{3} $ for x < 0
The Correct Option is D
Solution and Explanation
\ x + y = 2y, \\
x - y = 2y, \\
\end{array} \begin{array}
when y > 0 \\
when y < 0 \\
\end{array}$
$\Rightarrow \bigg \{ \begin{array}
\ y = x, \ when \ y > 0 \Rightarrow x > 0 \\
y = x/3, \ when \ y < 0 \Rightarrow x < 0 \\
\end{array}$
which could be plotted as,
Clearly, y is continuous for all x but not differentiable at x = 0,
Also, $ \frac{dy}{dx} = \bigg \{ \begin{array}
\ 1, \ x > 0 \\
1/3, \ x < 0 \\
\end{array} $
Thus, f (x) is defined for all x, continuous at x = 0,
differentiable for all x $ \in \ R - \{ 0 \}, \frac{dy}{dx}, = \frac{1}{3} $ for x < 0.
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Concepts Used:
Differentiability
Differentiability of a function A function f(x) is said to be differentiable at a point of its domain if it has a finite derivative at that point. Thus f(x) is differentiable at x = a
\(\frac{d y}{d x}=\lim _{h \rightarrow 0} \frac{f(a-h)-f(a)}{-h}=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}\)
⇒ f'(a – 0) = f'(a + 0)
⇒ left-hand derivative = right-hand derivative.
Thus function f is said to be differentiable if left hand derivative & right hand derivative both exist finitely and are equal.
If f(x) is differentiable then its graph must be smooth i.e. there should be no break or corner.
Note:
(i) Every differentiable function is necessarily continuous but every continuous function is not necessarily differentiable i.e. Differentiability ⇒ continuity but continuity ⇏ differentiability
(ii) For any curve y = f(x), if at any point \(\frac{d y}{d x}\) = 0 or does not exist then, the point is called “critical point”.
3. Differentiability in an interval
(a) A function fx) is said to be differentiable in an open interval (a, b), if it is differentiable at every point of the interval.
(b) A function f(x) is differentiable in a closed interval [a, b] if it is
- Differentiable at every point of interval (a, b)
- Right derivative exists at x = a
- Left derivative exists at x = b.