Calculus question?

Question

Consider the function

f(x)= -x^2-2x+1

Find the minimum on the interval [-2,2]

Answer 1

A minimum will either occur at the endpoints, or where f'(x) = 0.
f'(x) = -2x - 2, which is 0 only when x = -1.
So, our three possible points are x=-2, x = -1, x = 2.
Substituting each in give 1, 2, -7.
So the minimum was at x=2, with value -7.

Answer 2

f(x)= -x^2-2x+1

To find a minimum (or a maximum) you need to do a first derivative test. So let's find the first derivative!

f(x) = -x^2-2x+1
f'(x) = -2x-2

Now find where the function is either undefined or equal to zero.

0 = -2x-2 when x = -1

A zero happens when the first derivative changes from negative to positive (a min) or positive to negative (a max), which means that the original function is changing from decreasing to increasing (a min) or increasing to decreasing (a max).

We need to check how the first derivative is changing.

[-2, -1) -- first derivative is pos
(-1,2] -- first derivative is neg

This means that x = -1 is actually a maximum. But you need to find a minimum. But wait! Look at the interval [-2,2]! It is a closed interval because it uses [ ] instead of ( ), which includes the endpoints.

Plug these values, x = -2 and x = 2, into the original equation.
f(2) = -7
f(-2) = 1

The minimum on the interval is when x = 2, the minimum value is -7.

To make the process go faster you could have orignially just found f(2), f(-2), and f(-1) and picked the one that yielded the smallest value, but it never hurts to know the whole process. :)

Answer 3

DO NOT USE A CANNON TO KILL A BUTTERFLY

evaluating function at the end points would suffice to find the minimum b/c this parabola has ( a=-1 <0 ) branches pointing down so local minimum is out of question

Answer 4

f(x) = -x^2 - 2x + 1
f' = -2x -2 => x = -1

f(-1) = 2, which is the max
So given that it's an inverted parabola, the min is farthest away which is 2
f(2) = -7

Answer 5

f(x)=-x^2-2x+1
f'(x)=-2x-2
equating to 0
x=-1
f''(x)=-2
therefore it is a maximum
no min in the interval (-2,2)
additional information
it is an inverted parabola opening downwards
with the vertex at (-1,2)