I am given: $f, g: \mathbf{R^{2}} \to \mathbf{R} $ be differentiable functions. Fix $p \in \mathbf{R^{2}}$ such that $g(p) = 2020$. Assume that:
$$ || \bigtriangledown f || = 1 $$
and
$$ || \bigtriangledown g || = 1 $$
$$ \bigtriangledown f \cdot \bigtriangledown g = 1/2 $$
The question is can the function $f$ subject to the constraint $g(x,y) = 2020$ to attain the mininum.
My attempt: I claim that it cannot. I apply the use the Lagrange Multiplier. So the function $f$ subject to constraint $g(x,y) = 2020$ if $\bigtriangledown f$ is parallel to $\bigtriangledown g$.
But I use the formula of angle between $\bigtriangledown f, \bigtriangledown g$ for dot product:
$$ \theta = \cos^{-1}( \frac{ \bigtriangledown f \cdot \bigtriangledown g}{|| \bigtriangledown f || || \bigtriangledown g ||} ) = \cos^{-1}(1/2) $$ So $\bigtriangledown f$ is not parallel with $\bigtriangledown g$
Is my attempt to this problem plausible?