What are the local extrema, if any, of #f(x) =x^2 + 9x +1 #?
Parabolae have exactly one extrema, the vertex.
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
Using the power rule,
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.
so we have one critical point at
To find the y coordinate of the critical point we sub in
The critical point/vertex is
We know that because
To formally find if it's a maxima or minima we need to do the second derivative test.
The second derivative is 2 at all values of x. This means it is greater then zero everywhere, and the function is concave up everywhere (it's a parabola with