How does the gradient at (0,0) for for f=x*(y^2+10) point in direction of max increase?

Question

The gradient of $$f(x,y)=x(y^2+10)$$ is $[10,0]$. How does it point to the direction of fastest increase? By looking at the plot of the graph here we can see that the fastest change at $(0,0)$, would be along the direction of $[1,1]$ or $[1,-1]$ (the direction along the red flap.

Answer 1

Too long for an addition to Hans Lundmark's comment.

A birds-eye view on the field lines of all gradients gives you a much better intuition in this case.

import matplotlib.pyplot as plt
from numpy import linspace, meshgrid

lim = 10
x = linspace(-lim,lim,10)
y = linspace(-lim,lim,10)
X, Y = meshgrid(x,y)  
# Assign vector directions
Xp = Y**2+10
Yp = 2*X*Y
plt.streamplot(X,Y,Xp,Yp, density=1.4, linewidth=None)
plt.legend()
plt.show()