Question

Let $f$ and $g$ be two functions defined on an interval I such that $f(x) \geq 0$ and $g(x) \leq 0$ 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 $f(x) \geq 0$ and $g(x) \leq 0, x \in$. 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}{d x}[f(x) g(x)]=\underbrace{f^{\prime}(x) g(x)}_{\text {+ve }}+\underbrace{f(x) g^{\prime}(x)}_{\text {+ve }}$
$\Rightarrow \frac{d}{d x} f(g(x))=f^{\prime}(g(x)) \cdot g^{\prime}(x)<0$
$\Rightarrow (f o g)(x)$ is decreasing function