Question

Is `f(x)=(x+1)(x+5)(x-7)` increasing or decreasing at `x=-1`?

Answer 1

Decreasing

Explanation:

While you can figure out the answer by calculating the derivative `f^prime (-1)` the traditional way, the following approach (which works when you are trying to find the derivative of a function at one of its zeroes) is often quicker.

When `x` is close to -1, we have `x+5 ~~ -1+5 = 4` and `x-7 ~~ -1-7 = -8`. So, in the immediate vicinity of `-1`, the function is approximately

`f(x) ~~ (x+1) times 4 times (-8) = -32(x+1)`

Thus the function is decreasing.

In fact, this method also gives you the value of `f^prime(-1)` - it is easy to see that the value is -32.