With $f$ defined on the unit disc $D: x^2 +y^2 \leq 1$, and given that $\nabla f(1,0) = (1, 1)$, could $f$ reach its maximal value on $D$ at $(1, 0)$?

Question

$f$ is a continuous function with continuous partial derivatives.

With $f$ defined on the unit disc $D: x^2 +y^2 \leq 1$, and given that $\nabla f(1,0) = (1, 1)$, could $f$ reach its maximal value on $D$ at $(1, 0)$?

I know that the answer is no, but I am trying to prove how. I would love to share my input, but I'm at a complete loss as to what to do here. Any hint or suggestion would be welcome.

Answer 1

Hint: There are directions you can go from $(1,0)$ that stay inside $D$ and have positive dot product with $(1,1)$.