Let f and g be twice differentiable even functions on (- 2, 2) such...

Let $h$\left(x\right) = f\left(x\right)g ' \left(x\right)$roots<br />$∵ f$\left(x\right)$ is even $⇒ f\left(\dfrac{1}{4}\right) = f\left(\dfrac{1}{2}\right) = f\left(- \dfrac{1}{2}\right) = f\left(\dfrac{1}{4}\right) = 0 and$g\left(x\right)$has two roots as$g$\left(\dfrac{3}{4}\right) = g\left(\dfrac{- 3}{4}\right) = 0$ (as $g\left(x\right)$ is even function)

So, $h$\left(x\right) = f\left(x\right){g}^{'}\left(x\right)$will have$5$roots, hence its derivative${h}^{'}$\left(x\right) = f\left(x\right){g}^{"}\left(x\right) + {f}^{'}\left(x\right){g}^{'}\left(x\right)$ will have $4$ roots.