# What are the local extrema, if any, of #f(x) =x^2 + 9x +1 #?

Parabolae have exactly one extrema, the vertex.

It is

Since

You have two roots to finding the vertex of the parabola: one, use calculus to find were the derivative is zero; two, avoid calculus at all costs and just complete the square. We're going to use calculus for the practice.

By the linearity of the derivative we have

We set this equal to zero to find the critical points, the local and global minima and maxima and sometimes points of inflection have derivatives of zero.

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The function ( f(x) = x^2 + 9x + 1 ) has a local minimum at ( x = -4.5 ) and no local maximum.

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