Question

Let f(x) be a strictly increasing function such that f(x) < 0 for all values of x. Let g(x) be a strictly decreasing function.

Answer 1

when you multiply a less negative number by a positive number that is getting smaller, the product will become less negative. This means
`\( h(x) \)` is becoming less negative as
`\( x \)` increases, or in other words,
`\( h(x) \)` is increasing.

Therefore, the correct statement is:

C.
`\( h(x) < 0 \)` and is strictly increasing.

The given function
`\( h(x) = f(x)g(x) \)` is a product of two functions,
`\( f(x) \)` and
`\( g(x) \)`.

Here is what we know about these functions:

-
`\( f(x) \)` is strictly increasing and
`\( f(x) < 0 \)` for all values of
`\( x \)`.

-
`\( g(x) \)` is strictly decreasing and
`\( g(x) > 0 \)` for all values of
`\( x \)`.

Now, let's analyze the product
`\( h(x) = f(x)g(x) \)`:

- Since
`\( f(x) < 0 \)` and
`\( g(x) > 0 \)`, their product
`\( h(x) \)` will be negative for all
`\( x \)`. This means
`\( h(x) < 0 \)`.

- For the nature of
`\( h(x) \)` in terms of increasing or decreasing:

- When \( x \) increases,
`\( f(x) \)` increases because
`\( f \)` is strictly increasing.

- At the same time,
`\( g(x) \)` decreases because
`\( g \)` is strictly decreasing.

- The rate of increase of
`\( f(x) \)` and the rate of decrease of \( g(x) \) will determine the nature of
`\( h(x) \)`.

- However, without specific information about the rates of change of
`\( f(x) \)` and
`\( g(x) \)`, we cannot definitively say whether
`\( h(x) \)` is increasing or decreasing.

Here's what we can infer:

- Since
`\( f(x) \)` is negative and increasing, it is getting closer to zero from the negative side as
`\( x \)` increases, which makes
`\( f(x) \)` less negative.

- Since
`\( g(x) \)` is positive and decreasing, it is getting closer to zero as
`\( x \)` increases, but remains positive.

Now, when you multiply a less negative number by a positive number that is getting smaller, the product will become less negative. This means
`\( h(x) \)` is becoming less negative as
`\( x \)` increases, or in other words,
`\( h(x) \)` is increasing.

Therefore, the correct statement is:

C.
`\( h(x) < 0 \)` and is strictly increasing.

the complete Question is given below: