Calc f, f', f'' graphs

When f’(x) is > 0, above the x-axis

f(x) is increasing f’(x) to f(x)

When f’(x) is < 0, above the x-axis

f(x) is decreasing f’(x) to f(x)

When f’(x) = 0, on the x-axis

f(x) has a horizontal tangent (critical point) f’(x) to f(x)

f’(x) changes from positive to negative as it passes through the x-axis

f(x) has a local max (positive to negative)

f’(x) changes from negative to positive as it passes through the x-axis

f(x) has a local min (negative to positive)

If f’(x) = 0 but does not change sign

Inflection Point

When f’’(x) > 0, above the x-axis

f’(x) is increasing and f(x) is concave up

f(x) is concave up

f’’(x) is positive, f’’(x) > 0

f(x) is concave down

f’’(x) is negative, f’’(x) < 0

When f’’(x) < 0, below the x-axis

f’(x) is decreasing and f(x) is concave down

When f’’(x) = 0

f’(x) has a critical point (local max or min on the f’(x) graph)