## Conditions for maximum or maxima of a function

We can arrive at these conditions using the same approach as before.

Suppose, the function has a maximum at some point \( (c,d) \).

Since a maximum is a critical point, this means the gradient of the function is zero at \( (c,d) \).
Therefore, \(\dox{f}\bigg\rvert_{c,d} = 0 \) and \( \doy{f}\bigg\rvert_{c,d} = 0 \).

For the maximum or maxima to exist at \( (c,d) \), it should be the case that \( f(x+c,y+d) < f(c,d) \), for all \( (x,y) \in \real^2 \).

From the Taylor series expansion above, it also means,

$$ x^2 \doxx{f}\bigg\rvert_{c,d} + xy\doxy{f}\bigg\rvert_{c,d} + xy \doyx{f}\bigg\rvert_{c,d} + y^2 \doyy{f}\bigg\rvert_{c,d} < 0 $$

Using the same vector notation and Hessian notation as before, we note that the above inequality is equivalent to

$$ \vx^T \mH\bigg\rvert_{c,d} \vx < 0 $$

This means, \( \mH\bigg\rvert_{c,d} \) should be negative definite at the maximum.

Thus, a function has a maximum at a point \( (c,d) \) if

- Gradient at \( (c,d) \) is zero
- Hessian at \( (c,d) \) is negative definite