Question

Let $f$ and $g$ be two functions defined on an interval I such that $f(x)\ge 0$ and $g(x)\le 0$ for all $x\in I$ and $f$ is strictly decreasing on I while $g$ is strictly increasing on I then

• A. the product function $fg$ is strictly increasing on I
• B. the product function $fg$ is strictly decreasing on I
• C. fog $(x)$ is monotonically increasing on I
• D. All of these

Answer 1

Answer: (A)

Since $f(x)\ge 0$ and $g(x)\le 0,x\in$. Also $f(x)$ is stricity decreasing on $I$, therefore $f^{\prime }(x)<0$ and $g(x)$ is strictly increasing on $I$, therefore $g^{\prime }(x)>0$
Now, $\frac{d}{dx}[f(x)g(x)]=\underset{\text{+ve }}{\underbrace{f^{\prime }(x)g(x)}}+\underset{\text{+ve }}{\underbrace{f(x)g^{\prime }(x)}}$
$\Rightarrow \frac{d}{dx}f(g(x))=f^{\prime }(g(x))\cdot g^{\prime }(x)<0$
$\Rightarrow (fog)(x)$ is decreasing function