#### Transcript of Relationship between f, f', and f''

**F' to F(x)**

If f' is positive at x=c, then the slope of function f(x) at x=c is increasing

If f' is negative at x=c, then the slope of function f(x) at x=c is decreasing

If f' is zero at x=c, then the slope of function f(x) at x=c is zero and it is a critical point.

HW Problems

If this was a graph of f '(x)

On what intervals on the graph is f(x) concave up?

Where is f(x) increasing?

If this was a graph of f ''(x)

where is f(x) decreasing?

where is f '(x) positive?

F'' to F(x)

If f'' is positive at x=c, then the concavity of function f(x) at x=c is concave up

If f'' is negative at x=c, then the concavity of function f(x) at x=c is concave down

Inflation points ONLY occurs when f'' is zero at x=c AND when f(x) at x=c changes between concave up and concave down

**Relationship between f, f', and f''**

If function f(x) is a continuous function, then the derivation of f(x) is f'(x)

f(x) is the original function

f'(x) defines the slope of the tangent line

f''(x) defines the concavity (concave up or concave down) of the function f(x)

How to find f, f', and f''

f is the given function... to find, read

f' is found by finding the derivative of f (see group 6)

f'' is the derivative of f' (also see group 6)

To go backwards, use anti-differentiation methods (1/n+1)n^n+1)

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