How do you find the exact relative maximum and minimum of the polynomial function of y=x^2 +x -1y=x2+x1?

1 Answer
Jun 26, 2017

we have a minimum at (-1/2,-5/4)(12,54)

Explanation:

To find a min/max we look for values of xx that make the derivative vanish. To determine the nature of those turning points we perform the second derivative test

We are dealing with a positive quadratic so we expect a single minimum.

We have:

y = x^2 + x -1 y=x2+x1

Differentiating wrt xx we get:

dy/dx = 2x+1 dydx=2x+1

For the derivative to vanish we have:

dy/dx = 0 => 2x+1 =0 dydx=02x+1=0
:. x=-1/2

Differentiating again wrt x we get:

(d^2y)/(dx)^2 = 2

So when x=-1/2 => (d^2y)/(dx)^2 > 0 confirming a minimum.

Finally, When x=-1/2 => y= 1/4-1/2-1 = -5/4

Thus we have a minimum at (-1/2,-5/4)

graph{y = x^2 + x -1 [-10, 10, -5, 5]}