How derivatives affect the shape of a graph ( Section 4.3) Alex Karassev

First and second derivatives f ′ tells us about intervals of increase and decrease f ′′ tells us about concavity

First derivative: Intervals of Increase / Decrease

Increasing / Decreasing x y = f(x) slope < 0 slope > 0 Slope of tangent line decr. incr.

Increasing / Decreasing Test x y = f(x) f ′ (x) < 0 f ′ (x) > 0 Derivative decr. incr.

Change of behavior x y = f(x) f ′ (x) < 0 f ′ (x) > 0 f can change from increasing to decreasing and vice versa: at the points of local max/min (i.e. at the critical numbers) at the points where f is undefined decr. incr.

Local max/min: 1 st derivative test x f ′ (x) < 0 f ′ (x) > 0 decr. c loc. min. loc. max Let c be a critical number How do we determine whether it is loc. min or loc. max or neither? y = f(x) If f ′ changes from negative to positive at c, it is loc. min. If f ′ changes from positive to negative at c, it is loc. max. If f ′ does not change sign at c, it is neither (e.g. f(x) = x 3, c =0)

Second derivative: Concavity

Concavity: definition Graph lies above tangent lines: concave upward Graph lies below tangent lines: concave downward

Concavity: example y = f(x) up down up down up Inflection points

Concavity test: use f′′ Graph lies above tangent lines: concave upward Graph lies below tangent lines: concave downward f′′ (x) > 0 f′′ (x) < 0 Inflection points: Numbers c where f′′(c) = 0 are "suspicious" points

Change of concavity y = f(x) up down up down up f can change from concave upward to concave downward and vice versa: at inflection points (check f ′′ (x) = 0) at the points where f is undefined

Local max/min: 2 nd derivative test x f ′′ (c) > 0 f ′′ (c) < 0 c loc. min. loc. max Suppose f ′ (c) = 0 How do we determine whether it is loc. min or loc. max or neither? y = f(x) If f ′′ (c) > 0 the graph lies above the tangent ⇒ loc. min. If f ′′ (c) < 0 the graph lies below the tangent ⇒ loc. max. If f ′′ (c) = 0 the test is inconclusive (use 1 st deriv. test instead!) NOTE: tangent line at (c,f(c)) is horizontal

Comparison of 1 st and 2 nd derivative tests for local max/min Second derivative test is faster then 1 st derivative test (we need to determine where f′(c) = 0 and then just compute f′′(c) at each such c) Second derivative test can be generalized on the case of functions of several variables However, when f′′(c) = 0, the second derivative test is inconclusive (for example, (0,0) is an inflection point for f(x) = x 3, while for x 4 it is a point of local minimum, and for –x 4 it is a point of local maximum)

Examples Sketch the graph of function y = x 4 – 6x 2 Use the second derivative test to find points of local maximum and minimum of f(x) = x/(x 2 +4)