How do I find the minimum point for the equation y = x^2 -5x - 6?
The minimum point for the curve can be found by finding the point at which the gradient is zero, where the curve is instantaneously level, before increasing again. This can be done by differentiating the equation of the line, to give
dy/dx = 2x-5, the gradient of the curve.
Which is then solved for dy/dx = 0, to give x = 2.5
However, there is an extra step to show what kind of stationary point this is. dy/dx = 2x-5 must be further differentiated, to find the rate of change of the gradient, d2y/dx2, and evaluated at x = 2.5. In this case, giving d2y/dx2 = 2. If this is positive, as in this case, the gradient is increasing, so this must be a minimum point. Vice versa, if this is negative, the gradient is decreasing, and therefore this must be a maximum point. To find the minimum value of y, the original equation can then be evaluated at x = 2.5, giving y = -12.25
