Answer
All the points on the line $2x+y=0$ are the points where the tangent line is horizontal.
Work Step by Step
Differentiate $3x^2 + 4y^2 + 3xy = 24$ with respect to $x$ using product rule and chain rule.
We get, $3\times2x+4\times2y\dfrac{dy}{dx}+3y\dfrac{dx}{dx}+3x\dfrac{dy}{dx}=0$
$\implies 6x+8y\dfrac{dy}{dx}+3y+3x\dfrac{dy}{dx}=0$
If the tangent line is horizontal, then $\dfrac{dy}{dx}=0$.
Therefore, we get $6x+3y=0$ or $2x+y=0$.
Hence, all the points on the line $2x+y=0$ are the points where the tangent line is horizontal.
