Find the x-Coordinate of the minimum point of the function: f(x) = x^2 + 2x - 5

One of the easiest ways to find a minimum/ maximum point is using differentiation.

The derivative of the function f(x) is usually written as dy/dx or f'(x) and is essentially a function that tells us the gradient of f(x) at any point x.

One of the key characteristics of a minimum/ maximum of a function is the way the gradient behaves at & around it.

For a minimum point (say m), the function is decreasing before and increasing after m. i.e. it as a negative gradient (f'(x) < 0) to the left of m and a positive gradient (f'(x) > 0) to the right of m (This is reversed for a maximum). In order for the sign of the gradient to change, it has to pass through 0, and this is exactly where m is. i.e. f'(x) = 0.

So if differentiate f(x), we get f'(x) = 2x + 2 and we want to set f'(x) = 0 to find the x value of the minimum, m.

f'(x) = 0 = 2x +2 2x + 2 = 0 2x = -2 (subtract 2 on both sides) x = -1 (divide by 2 on both sides)

And so we have the x-coordinate of the minimum at x = -1; we can check that this is indeed a minimum by taking a point to the left of -1 e.g. -2 and a point to the right e.g. 0. by inputting these points into f'(x) and checking their signs we can confirm the behaviour of f(x) around x = -1.

f'(-2) = -2 < 0 and f'(0) = 2 > 0 - which confirms that the gradient is indeed negative before m and positive after and so m ( x= -1) is the minimum of f(x).