Chapter 5.2, Problem 73E

To determine

Whether the function is even, odd, neither.

Answer to Problem 73E

The function $f\left(x\right)=x^2\sin x$ is an odd function

Explanation of Solution

Given information:

  $f\left(x\right)=x^2\sin x$

Formula used:

  $\begin{array}{l}f\left(x\right)=f\left(-x\right):\text{ Condition for even function}\\ \\ f\left(x\right)=-f\left(x\right):\text{ Condition for odd function}\\ \\ \sin \left(-\theta \right)=-\sin \theta \end{array}$

  $f\left(x\right)=x^2\sin x$

Now we know that if we put $x=-x$ in the equation and it results into $f\left(x\right)=f\left(-x\right)$ the equation is “Even”, and if the equation results into $f\left(x\right)=-f\left(x\right)$ the equation is odd.

Therefore, putting value of $x=-x$ in the equation to find the type of the equation:

  $\begin{array}{l}f\left(-x\right)={\left(-x\right)}^2.\sin \left(-x\right)\\ f(-x)=x^2.\left(-\sin x\right)\because \sin \left(-\theta \right)=-\sin \theta \\ f\left(-x\right)=-x^2\sin x\\ \therefore f\left(-x\right)=-f\left(x\right)\end{array}$

Hence, the function

  $f\left(x\right)=x^2\sin x$ is an odd function.