Estimate the local minimum of y=−x3−5x^2−3x+9.

Answer 1

take the derivative and set =0

-3x^2 - 10x -3 = 0

try to factor or use the quadratic formula

3x^2 +10x + 3 = 0

notice 3x3 + 1 = 10

(3x+1)(x+3) = 0

local extrema are x =-3 and x=-1/3

one of them is a local minimum.

plug those values into the cubic equation to see which.

x=-3 gives zero

while

x=-1/3 gives almost + 10

x=-3 is the local minimum or the point (-3,0)

Answer 2

The minimum occurs at one of the places where 3x²+10x+3 = 0.

You can tell which one using the 2nd derivative test or by graphing.