Question

Which interval contains a local minimum for the grafted function? [-4, -2.5] [-2, -1] [1, 2] [2.5, 4]

Answer 1

Last option [2.5, 4]

Explanation:

The local minima of a function are the points where the slope of the curve is zero and the function reaches a minimum value.

If, on the contrary, the function reaches a maximum value, then that point is a local maximum.

Observe, for example, the point (3, -4). Note that a line tangent to that point would be horizontal, that is, of slope m = 0. This point is a minimum of the function, because there are no values x to the right x = 3 or to the left of x = 3 for which
`f(x) <-4`

Now we must search among the options that interval contains at this point.

The intervals [-4, -2.5] [-2, -1] do not contain any local minimum

The intervalor [1, 2] contains a local maximum.

The only interval that contains a local minimum is [2.5, 4], which contains the point (3, -4)