1. ## Gradient, Function 2 variables

Hi everybody. Given is a function f(x,y). Sought is a point Px/Py on f(x,y)=1, whose tangent is perpendicular to the vektor n=(1,0). Without any reference to the specific function in the solution it is stated, that grad(Px/Py)=\lambda*n=1,0.

1) why is this generally true?

(If needed, the specific function is f(x,y)=3{x}^{2} +xln(y)+{e}^{ax} , x1≥0,x2>0,a∈R.)
2. grad f always points in the direction of fastest increase. Further, the directional derivative, the derivative in the direction of unit vector v, is the dot product $\displaystyle \nabla f\cdot v$. From that it follows that if the rate of change in the direction of vector v is 0, $\displaystyle \nabla f\cdot v= 0$ which says that v is perpendicular to $\displaystyle \nabla f$.
In particular, along the curve f(x,y)= 1 (or any constant) the value of f does not change so its derivative along that curve (in the direction tangent to the curve) is 0. That is, $\displaystyle \nabla f\cdot v= 0$ for v tangent to the curve so $\displaystyle \nabla f$ is perpendicular to the curve.