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. They are also part of precalculus. They find their applications in fields other than mathematics.
Answer: A function that is positive in the entire interval [-3, -2] is -x2 - 5x - 5.
Let's understand the solution.
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 has a downward parabola. Hence:
⇒ f(x) = - (x + 3) (x + 2) = - x2 - 5x - 6
Hence, we see that the above function found out 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 than one functions which are positive in [-3, -2] using various methods. This solution is not unique.