7.5 Turning points

AQA Edexcel OCR A OCR B (MEI)
Stationary points

Stationary points occur when dydx=0\dfrac{dy}{dx}=0. The second derivative can be used to test whether the stationary point is a maximum or a minimuum.

Maximum points have dydx=0\dfrac{dy}{dx}=0 and d2ydx2<0\dfrac{d^2y}{dx^2} < 0.


Minimum points have dydx=0\dfrac{dy}{dx}=0 and d2ydx2>0\dfrac{d^2y}{dx^2} > 0.



Points of inflection

Points of inflection occur when d2ydx2=0\dfrac{d^2y}{dx^2} = 0. Points of inflections can be stationary or non-stationary.

Stationary points of inflection have dydx=0\dfrac{dy}{dx}=0 and d2ydx2=0\dfrac{d^2y}{dx^2} = 0.


Non-stationary points of inflection have dydx≠0\dfrac{dy}{dx} \neq 0 and d2ydx2=0\dfrac{d^2y}{dx^2} = 0.



Decreasing or increasing functions

Sections of curves can be decreasing or increasing.

Decreasing sections of curves have dydx<0\dfrac{dy}{dx} < 0.


Increasing sections of curves have dydx>0\dfrac{dy}{dx} > 0.



Concave or convex

Sections of curves can be concave (n-shaped, imagine the entrance of a cave) or convex (u-shaped).

Concave sections of curves have d2ydx2<0\dfrac{d^2y}{dx^2} < 0.


Convex sections of curves have d2ydx2>0\dfrac{d^2y}{dx^2} > 0.


Important
Stationary points occur when dydx=0\dfrac{dy}{dx}=0.

Maximum points have dydx=0\dfrac{dy}{dx}=0 and d2ydx2<0\dfrac{d^2y}{dx^2} < 0.

Minimum points have dydx=0\dfrac{dy}{dx}=0 and d2ydx2>0\dfrac{d^2y}{dx^2} > 0.

Stationary points of inflection have dydx=0\dfrac{dy}{dx}=0 and d2ydx2=0\dfrac{d^2y}{dx^2} = 0.

Non-stationary points of inflection have dydx≠0\dfrac{dy}{dx} \neq 0 and d2ydx2=0\dfrac{d^2y}{dx^2} = 0.

Decreasing sections of curves have dydx<0\dfrac{dy}{dx} < 0 and increasing sections of curves have dydx>0\dfrac{dy}{dx} > 0.

Concave sections of curves have d2ydx2<0\dfrac{d^2y}{dx^2} < 0 and convex sections of curves have d2ydx2>0\dfrac{d^2y}{dx^2} > 0.
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