Give a function that is positive for the entire interval [–3, –2] and is represented by a downward parabola.

Functions are one of the main concepts used in mathematics. A positive function in an interval implies that the range of the function is greater than 0 in that interval.

Answer: A function that is positive in the entire interval [-3, -2] is -x² - 5x - 5.

Let's understand the solution.

Explanation:

If the function is represented by a downward parabola, then the function must be quadratic and the coefficient of x² must be negative.

Now, to find the required function, let us assume that -3 and -2 are the roots of the equation and represented by a downward parabola. 

⇒ f(x) = - (x + 3) (x + 2) = - x² - 5x - 6

Hence, we see that the above function has a value of zero at x = -3 and x = -2.  But, we need a positive value at both ends too. Hence, let's check the graph for - x² - 5x - 5.

In the graph above, we see that the graph is positive in [-3, -2]. Hence, the function -x² - 5x - 5 is positive in [-3, -2].

Note that you can find more functions that are positive in [-3, -2] using various methods. This solution is not unique.

Hence, a function that is positive in the entire interval [-3, -2] and represented by a downward parabola is -x² - 5x - 5.