If f(x)= sin ((π/3)[x]-x2), where 2

Question

If f(x)= sin ((π/3)[x]-x2), where 2< x< 3 and [.] represents greatest integer function, then f'(√(5 π/3)) is equal to (A) √(5 π/3) (B) 2 √(5 π/3) (C) -2 √(5 π/3) (D) -√(5 π/3)

Tardigrade Admin · Accepted Answer

Since, 2< x< 3, ∴ [x]=2 ∴ f(x)= sin ((2 π/3)-x2) ∴ f'(x)= cos ((2 π/3)-x2)(-2 x) ∴ f'(√(5 π/3))=2 √(5 π/3)

Q. If $f (x) = sin (\frac{π}{3} [x] - x^{2})$ , where $2 < x < 3$ and $[.]$ represents greatest integer function, then $f^{'} (\frac{5 π}{3})$ is equal to

$\frac{5 π}{3}$

$2 \frac{5 π}{3}$

$- 2 \frac{5 π}{3}$

$- \frac{5 π}{3}$

Since, $2 < x < 3,$
$∴ [x] = 2$
$∴ f (x) = sin (\frac{2 π}{3} - x^{2})$
$∴ f^{'} (x) = cos (\frac{2 π}{3} - x^{2}) (- 2 x)$
$∴ f^{'} (\frac{5 π}{3}) = 2 \frac{5 π}{3}$

Q. If f(x)=sin(3π​[x]−x2), where 2<x<3 and [.] represents greatest integer function, then f′(35π​​) is equal to

35π​​

235π​​

−235π​​

−35π​​