4 mistakes learners make with stationary points, each one named and explained.
Set the first derivative to zero to find them, and use the second to say what they are.
A point of inflection or a saddle point is chosen where a clear maximum or minimum exists.
The list of possible natures is longer than the ones that usually occur.
Question. Second derivative is 6, so not zero
A common answer. a point of inflection
The answer. a local minimum
Why. A point of inflection needs the second derivative to be zero there.
Work out the second derivative's value first. Zero or not zero decides which list you are choosing from.
The derivative is correct and the quadratic that follows is factorised incorrectly.
The calculus is done and attention drops for what looks like routine algebra.
Question. 3x^2 - 12x + 9 = 0
A common answer. x = 0 and x = 4
The answer. x = 1 and x = 3
Why. Dividing by 3 gives x^2 - 4x + 3, which factorises to (x - 1)(x - 3).
Take out the common factor first, then factorise. Substituting back checks it in seconds.
The second derivative is used where the first was needed, giving the point of inflection instead.
Both derivatives get set to zero in this topic, for two different purposes.
Question. y = x^3 - 6x^2 + 9x + 1
A common answer. 6x - 12 = 0, so x = 2
The answer. 3x^2 - 12x + 9 = 0, so x = 1 and 3
Why. Stationary points are where the gradient is zero, and the gradient is the first derivative.
Ask what is zero at the point you want. A flat gradient means the first derivative.
A positive second derivative is called a maximum.
Both are turning points and which sign means which is a bare convention.
Question. y = x^3 - 3x at x = 1, second derivative 6
A common answer. a local maximum
The answer. a local minimum
Why. A positive second derivative means the curve is bending upwards, which is a minimum.
Draw the smile and the frown. A smile holds water and is positive.
WAJD spots these patterns in your child's answers and names the one behind their wrong answers, instead of just marking them wrong.