# How do I write vertex as a maximum or minimum?

Question asks me to label the "vertex as a maximum or minimum"

The function is:f(x) = x^2+x-6

I found the vertex to be:

(-1/2,-25/4)

And it's at the bottom of the graph, not at the top so it would be minimum.

How do I use this information to answer this question?

Do I just say the vertex of the function is (-1/2,-25/4), which is the minimum of the graph?

Thanks! Relevance
• f(x) = x² + x - 6

or, f(x) = (x + 1/2)² - 25/4

Hence, vertex is at (-1/2, -25/4)

Now, as the coefficient of x² is positive, the function (parabola) will be ∪ - shaped

so, (-1/2, -25/4) is labelled as a minimum

A sketch is below.

:)> • If the term in 'x^2' is positive(+), then the vertex is a minimum; curve is bowl shaped.

If the term in 'x^2' is negative(-), then the curve is a maximum,  curve is umbrella shaped.

You can doubly differentiate, then if the answer is positive/negative(+/-), then it is a minimum/maximum.

Hence

f'(x) = 2x + 1

f''(x) = (+)2 so it is a minimum.

• If you look at the graph, there are NO values lower than -25/4, and all the rest are higher.

That's the definition of a minimum -- the lowest value in the given range (which can be a local minimum)