You need to express x and y in terms of z first.

Then evaluate yz+xz in terms of z.

You get a quadratic in z, with the co-efficient of the square being negative.

Hence the quadratic in z is an inverted U-shape.

The derivative of the quadratic being zero (horizontal tangent resting on top of the curve) locates z giving max yz+xz.

Then

Evaluate this and differentiate (then set derivative to zero) to find z that gives the maximum yz+xz.

Substitute in this value of z to find the maximum value of yz+xz.