Question

Let $f: R \rightarrow R$ be a function defined by $f(x)=\sin \pi x(|x-1||x-2||x+1||x-3|)$, 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

Answer 1

Answer: (B), (C), (D)

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