If f (3)(p) ≠ 0, then f has an inflection point at p. Otherwise, if f (4)(p) ≠ 0, then f has a local minimum at p if f (4)(p) > 0 and a local maximum if f (4)(p) < 0.

1) If f'(x) > 0 for all x on (a,c) and f'(x)<0 for all x on (c,b), then f(c) is a local maximum value. 2) If f'(x) < 0 for all x on (a,c) and f'(x)>0 for all x on (c,b), then f(c) is a local maximum value. 3) If f'(x) has the …

A function f has a local maximum at c if there exists an open interval I containing c such that I is contained in the domain of f and f(c) ≥ f(x) for all x ∈ I.

Characterization of local extrema Theorem (First Derivative Test) If a differentiable function f has a local maximum or minimum at (a, b) then holds ∇f = h0, 0i. (a,b) Remark: The tangent plane at a local …

Imagine you are standing on a local maximum of the graph of a two-variable function z = f(x; y). This means that in every direction you travel, you will lose elevation.

Find the critical points for each function. Use the first derivative test to determine whether the critical point is a local maximum, local minimum, or neither.

The point at which the derivative should cross zero (or fail to exist) is c, the local maximum. Similar analysis would indicate that the derivative should be zero (or not exist) at a local minimum, but with …