Number of points where fx-x-sin2- x is not differentiable if x-7-10 is where - denotes the greatest integer function

F(x)=[x]sin^2(π x) f(x)= 7sin^2(π x), amp;x ∈( 7, 6) 0, amp; x= 6 6sin^2(π x), amp;x∈( 6, 5) 0, amp;x= 5 … amp;… … nbsp; amp;… … nbsp; ...

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Standard XII

Mathematics

Right Hand Derivative

Number of poi...

Question

Number of points where $f (x) = [x] {sin}^{2} (π x)$ is not differentiable if $x \in (- 7, 10)$ is (where $[.]$ denotes the greatest integer function)

15

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0

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6

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17

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Solution

The correct option is B $0$
$f (x) = [x] {sin}^{2} (π x)$
$f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} - 7 {sin}^{2} (π x), & x \in (- 7, - 6) 0, & x = - 6 - 6 {sin}^{2} (π x), & x \in (- 6, - 5) 0, & x = - 5 \dots & \dots \dots & \dots \dots & \dots 0, & x = 9 9 {sin}^{2} (π x), & x \in (9, 10) \end{matrix}$
Clearly $f (x)$ is continuous for $x \in (- 7, 10)$

$f^{'} (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} - 7 sin (2 π x), & x \in (- 7, - 6) 0, & x = - 6 - 6 sin (2 π x), & x \in (- 6, - 5) 0, & x = - 5 \dots & \dots \dots & \dots \dots & \dots 0, & x = 9 9 sin (2 π x), & x \in (9, 10) \end{matrix}$
Clearly $f (x)$ is differentiable for $x \in (- 7, 10)$
Hence, there is no such point where $f (x)$ is not differentiable for $x \in (- 7, 10)$

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Number of points where f(x)=[x]sin2(πx) is not differentiable if x∈(−7,10) is (where [.] denotes the greatest integer function)

Number of points where $f (x) = [x] {sin}^{2} (π x)$ is not differentiable if $x \in (- 7, 10)$ is (where $[.]$ denotes the greatest integer function)