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Integration of Piecewise Continuous Functions

Let fx=|x-a...

Question

Let $f (x) = | x - a | ϕ (x)$ where $ϕ$ is a continuous function and $ϕ (a) \neq 0$ Then-

A

f^{'} (a^{+}) = ϕ^{'} (a)

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B

f is differentiable at

x = a

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C

f^{'} (a) = ϕ^{'} (a)

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D

f^{'} (a^{-}) = - ϕ (a)

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Solution

The correct option is D $f^{'} (a^{-}) = - ϕ (a)$
$f (x) = {\begin{matrix} (a - x) ϕ (x) i f x < a (x - a) ϕ (x) i f x > a \end{matrix}$
$f^{'} (x) = {\begin{matrix} (a - x) ϕ^{'} (x) - ϕ (x) i f x < a (x - a) ϕ^{'} (x) + ϕ (x) i f x > a \end{matrix}$
Hence $f^{'} (a -) = - ϕ (a), f^{'} (a +) = ϕ (a)$

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Similar questions

f (x) = x^{2}, ϕ (a) = b^{2}, f^{'} (a) = n ϕ^{'} (a)

ϕ^{'} (a) \neq 0

lim x \to a (\frac{\sqrt{f (x)} - a}{\sqrt{ϕ (x)} - b})

Q. If f (x) = | x − a | ϕ (x), where ϕ (x) is continuous function, then
(a) f' (a⁺) = ϕ (a)
(b) f' (a⁻) = −ϕ (a)
(c) f' (a⁺) = f' (a⁻)
(d) none of these

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3 ϕ^{'} (a)

lim x \to 0 \frac{\sqrt{f (x)} - a}{\sqrt{ϕ (x)} - b}

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lim x \to 0 \frac{\sqrt{f (x)} - a}{\sqrt{ϕ (x)} - b}

Q. Match the following lists Let f(x) be any function

List - I	List - II
A) $f^{'} (a) = 0$ and $f^{''} (a) < 0$ then	l) f(x) is increasing at $x = a$
B) $f^{'} (a) = 0$ and $f^{''} (a) > 0$ then	2) f(x) has maximum value at $x = a$
C) $f^{'} (a) \neq 0$ then	3) f(x) has neither maximum nor minimum
D) $f^{'} (a) > 0$	4) f(x) has minimum value at $x = a$
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