Let f: R arrow R be a function defined by f(x)= sin π x(|x-1||x-2||x+1||x-3|), then which of the following is'are correct?

Question

Let f: R arrow R be a function defined by f(x)= sin π x(|x-1||x-2||x+1||x-3|), then which of the following is'are correct? (A) y=f(x) is continuous but not differentiable (B)  y=f(x) is differentiable (C) f prime prime(x)=0 has at least 3 distinct real roots (D) y=f(x) is an onto function

Tardigrade Admin · Accepted Answer

f ( x )= sin π x (| x +1|| x -1|| x -2|| x -3|) at x=-1,1,2,3 the function is differentiable and f prime(-1)=f prime(1)=f prime(2)=f prime(3)=0 ⇒ From rolle's theorem f prime prime( x )=0 has at least 3 real roots.

Q. Let $f : R \to R$ be a function defined by $f (x) = sin π x (∣ x - 1∣∣ x - 2∣∣ x + 1∣∣ x - 3∣)$ , then which of the following is'are correct?

$y = f (x)$ is continuous but not differentiable

$y = f (x)$ is differentiable

$f^{''} (x) = 0$ has at least 3 distinct real roots

$y = f (x)$ is an onto function

$f (x) = sin π x (∣ x + 1∣∣ x - 1∣∣ x - 2∣∣ x - 3∣)$
at $x = - 1, 1, 2, 3$ the function is differentiable and $f^{'} (- 1) = f^{'} (1) = f^{'} (2) = f^{'} (3) = 0$
$\Rightarrow$ From rolle's theorem $f^{''} (x) = 0$ has at least 3 real roots.

Q. Let f:R→R be a function defined by f(x)=sinπx(∣x−1∣∣x−2∣∣x+1∣∣x−3∣), then which of the following is'are correct?

y=f(x) is continuous but not differentiable

y=f(x) is differentiable

f′′(x)=0 has at least 3 distinct real roots

y=f(x) is an onto function