5CUsing2ndDerivatives

Sketching with Second Derivatives

We know that for the __first__ derivative:

Recall that a ** point of inflection ** is a point where the curve changes between concave and convex (or concave up and concave down). A point of inflection can be a stationary point but does not have to be.

Example

math \text {For } f(x)=x^3-2x^2-5x+6, \; \text {Sketch } f(x), f'(x) math

math f'(x)= 3x^2-4x-5 math

Second Derivative



Example

math \text {For } f(x)=x^3-2x^2-5x+6, \; \text {Sketch } f(x), f''(x) math

math \\ f'(x)= 3x^2-4x-5 \\ f''(x) = 6x - 4 math



NOTE: To have a point of inflection, the second derivative must CROSS the x-axis, not just touch it. The second derivative must change sign between positive and negative.

Summary

Combining information about first and second derivatives into one table, we get:

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