Stationary Points & Turning Points Flashcards

All 7 cards in this deck

What is a stationary point on a curve?

A point where $\frac{dy}{dx} = 0$ (the gradient is zero).

What are the steps to find the coordinates of the stationary points of a curve?
e.g. $y = x^2 - 4x + 1$

1. Differentiate: $\frac{dy}{dx} = 2x - 4$
2. Set $\frac{dy}{dx} = 0$ and solve: $x = 2$
3. Substitute into the curve equation for $y$: $(2, -3)$

How does the second derivative tell you the nature of a stationary point?

$\frac{d^2y}{dx^2} < 0$ gives a maximum; $\frac{d^2y}{dx^2} > 0$ gives a minimum.

How does the sign of the gradient either side of a stationary point tell you its nature?

$\frac{dy}{dx} > 0$ then $< 0$ gives a maximum; $\frac{dy}{dx} < 0$ then $> 0$ gives a minimum.

What are the steps to find an unknown coefficient from a given stationary point?
e.g. $y = x^3 + bx^2$ has a stationary point at $x = 2$

1. Differentiate: $\frac{dy}{dx} = 3x^2 + 2bx$
2. Put $\frac{dy}{dx} = 0$ at the given $x$: $12 + 4b = 0$
3. Solve: $b = -3$

True or false? At a maximum point on a curve the tangent is parallel to the $x$-axis.

True. The gradient there is zero.

On a sketch, what happens to the curve between a minimum point and the maximum point that follows it (as $x$ increases)?

The function is increasing (gradient positive).