7.5 Turning points
Stationary points
Stationary points occur when \(\dfrac{dy}{dx}=0\). The second derivative can be used to test whether the stationary point is a maximum or a minimuum.
Points of inflection
Points of inflection occur when \(\dfrac{d^2y}{dx^2} = 0\). Points of inflections can be stationary or non-stationary.
Decreasing or increasing functions
Sections of curves can be decreasing or increasing.
Concave or convex
Sections of curves can be concave (n-shaped, imagine the entrance of a cave) or convex (u-shaped).
Stationary points occur when \(\dfrac{dy}{dx}=0\). The second derivative can be used to test whether the stationary point is a maximum or a minimuum.
Maximum points have \(\dfrac{dy}{dx}=0\) and \(\dfrac{d^2y}{dx^2} < 0\).
Minimum points have \(\dfrac{dy}{dx}=0\) and \(\dfrac{d^2y}{dx^2} > 0\).
Points of inflection
Points of inflection occur when \(\dfrac{d^2y}{dx^2} = 0\). Points of inflections can be stationary or non-stationary.
Stationary points of inflection have \(\dfrac{dy}{dx}=0\) and \(\dfrac{d^2y}{dx^2} = 0\).
Non-stationary points of inflection have \(\dfrac{dy}{dx} \neq 0\) and \(\dfrac{d^2y}{dx^2} = 0\).
Decreasing or increasing functions
Sections of curves can be decreasing or increasing.
Decreasing sections of curves have \(\dfrac{dy}{dx} < 0\).
Increasing sections of curves have \(\dfrac{dy}{dx} > 0\).
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 \(\dfrac{d^2y}{dx^2} < 0\).
Convex sections of curves have \(\dfrac{d^2y}{dx^2} > 0\).
Important
Stationary points occur when \(\dfrac{dy}{dx}=0\).
Maximum points have \(\dfrac{dy}{dx}=0\) and \(\dfrac{d^2y}{dx^2} < 0\).
Minimum points have \(\dfrac{dy}{dx}=0\) and \(\dfrac{d^2y}{dx^2} > 0\).
Stationary points of inflection have \(\dfrac{dy}{dx}=0\) and \(\dfrac{d^2y}{dx^2} = 0\).
Non-stationary points of inflection have \(\dfrac{dy}{dx} \neq 0\) and \(\dfrac{d^2y}{dx^2} = 0\).
Decreasing sections of curves have \(\dfrac{dy}{dx} < 0\) and increasing sections of curves have \(\dfrac{dy}{dx} > 0\).
Concave sections of curves have \(\dfrac{d^2y}{dx^2} < 0\) and convex sections of curves have \(\dfrac{d^2y}{dx^2} > 0\).
Maximum points have \(\dfrac{dy}{dx}=0\) and \(\dfrac{d^2y}{dx^2} < 0\).
Minimum points have \(\dfrac{dy}{dx}=0\) and \(\dfrac{d^2y}{dx^2} > 0\).
Stationary points of inflection have \(\dfrac{dy}{dx}=0\) and \(\dfrac{d^2y}{dx^2} = 0\).
Non-stationary points of inflection have \(\dfrac{dy}{dx} \neq 0\) and \(\dfrac{d^2y}{dx^2} = 0\).
Decreasing sections of curves have \(\dfrac{dy}{dx} < 0\) and increasing sections of curves have \(\dfrac{dy}{dx} > 0\).
Concave sections of curves have \(\dfrac{d^2y}{dx^2} < 0\) and convex sections of curves have \(\dfrac{d^2y}{dx^2} > 0\).
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