How do you know if a stationary point on a curve is a maximum or minimum without plotting the graph?
Once you have found the stationary point of the equation by differentiating the equation and equating it to zero. You can then find out if the point you have found is a maximum, minimum or point of inflection by taking the differential equation you found. This gives you a second differential which shows you whether the gradient of the line is increasing or decreasing (and how fast) at a point . Then you plug in the value from the stationary point you found and the sign of the answer tells you the nature of it; if it's positive its a minimum, if it's negative its a maximum and if it equals zero its a point of inflection.Example:Given dy/dx = 3x2 + 6x , find the nature of the turning point at x=-2.Work out dy2/d2x = 6x+6Then plug in x=-2 to get: dy2/d2x = -12+6 =-6, therefore the turning point is a maximum as the second derivative is less than zero.
