Mathematics - Application of Derivative Question with Solution | TestHub

Let $h(x)=f(g(x))$.

$$h^{\prime }(x)=f^{\prime }(g(x))g^{\prime }(x)<0\forall x\in [0,\infty )$$

because $g^{\prime }(x)<0\forall x\in [0,\infty )$ and $f^{\prime }(g(x))>0\forall x\in [0,\infty )$.

Also, $h(0)=0$, and hence, $h(x)<0\forall x\in [0,\infty )$.

Let $p(x)=h(x^3-2x^2+2x)-h(4)$.

$$p^{\prime }(x)=h^{\prime }(x^3-2x^2+2x)\cdot (3x^2-4x+2)<0\forall x\in (0,2)$$

because $h^{\prime }(x^3-2x^2+2x)<0\forall x\in (0,\infty )$ and $3x^2-4x+2>0\forall x\in \mathbb{R}$.

This implies that $p(x)$ is a decreasing function.

$$\Rightarrow p(2)<p(x)<p(0)\forall x\in (0,2)$$

$$\Rightarrow h(4)-h(4)<p(x)<h(0)-h(4)$$

$$\Rightarrow 0<p(x)<-h(4)$$