Question

Find the equation of the tangent line to a curve y=-x²-1 that is parallel to the line 2x+y=6.

Answer 1

equation of the tangent of the curve at the point a can be found the following way

$f(x)-f(a)=f'(a)(x-a)$ , f'(a) is the slope of this tangent line, so, since this tangent line is parallel to the line 2x + y = 6, then f'(a) = -2

$y'=-2x\implies f'(a)=-2a\implies -2a=-2\implies a=1$

$f(1)=-2$

so, the sought equation is $y+2=-2(x-1)\implies y=-2x$