Question

Let y=x²+x,then mimimum value of y is

Answer 1

y = x²+x = (x+1/2)²-1/4

This is a parabola. At x= -1/2, y= -1/4.

If you compare it with y= x² and see that here at x=0,y=0, which is the minimum value. Similarly, if you shift this below the x axis by 1/2 unit and shift it to left of the origin by 1/4, you will get the same curve as stated in the question. So to find the minimum​ value of the the parabola, you just need to find its value at the vertex.

Hope, you understand. Good luck!
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Answer 2

y = x² + x

dy / dx = 2x + 1

• For maxima or minima put dy/dx = 0

2x + 1 = 0

x = -1/2

d²y/dx² = 2 + ve

• Minima At

x = -1/2

Minimum value = (-1/2)² + (-1/2)

⠀⠀ ⠀⠀ = 1/4 -1/2

⠀⠀⠀ ⠀⠀⠀ = (1–2)/4

⠀⠀⠀ ⠀⠀ ⠀⠀= -1/4