Q. 905.0( 1 Vote )

# Let f(x) = |sinx|. Then

A. f is everywhere differentiable

B. f is everywhere continuous but not differentiable

C. f is everywhere continuous but not differentiable at x = (2n + 1)

D. None of these

Answer :

Given that, f(x) = |sinx|

Let g(x) = sinx and h(x) = |x|

Then, f(x) = hog(x)

We know that, modulus function and sine function are continuous everywhere.

Since, composition of two continuous functions is a continuous function.

Hence, f(x) = hog(x) is continuous everywhere.

Now, v(x)=|x| is not differentiable at x=0.

Lv’(0) =

=

= (∵ v(x) = |x|)

=

=

=

Rv’(0) =

=

= (∵ v(x) = |x|)

=

=

=

⇒ Lv’ (0) ≠ Rv’(0)

⇒ |x| is not differentiable at x=0.

⇒ h(x) is not differentiable at x=0.

So, f(x) is not differentiable where sinx = 0

We know that sinx=0 at x = nπ, n ∈ Z

Hence, f(x) is everywhere continuous but not differentiable x = nπ, n ∈ Z

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