math upper homework-Real analysis due 9 hours later

MATH 105A - Fall 2018 - François Monard - UC Santa Cruz 92

Lecture 22 - 11/26 - Second derivatives.

Given f : [a,b] → R of class C1 on (a,b), we have seen that f ′ exists and is continuous on (a,b). If f ′ is in turn differentiable at x0 ∈ (a,b) (in the sense that limh→0(f ′(x0 + h) − f ′(x0))/h) exists, we say that f is twice differentiable at x0 and denote f

′′(x0) = (f ′)′(x0) (also called

d2f dx2

(x0), or second-order derivative of f at x0). If f is twice differentiable at every point of (a,b) and f

′′(x) is continuous on (a,b), we say that f is twice continuously differentiable on (a,b) (or, of class C2 on (a,b)).

The second derivative f ′′ is useful for: (i) finding local minima or maxima; (ii) determining the concavity of the graph of f.

Theorem 85 (Local properties of f ′′). Suppose f is differentiable on a neighbordhood of x0, and suppose f ′′(x0) exists. Define g(x) := f(x0) + f

′(x0)(x − x0) (the best linear approximation to f near x0).

(a) Suppose f ′(x0) = 0. If f ′′(x0) > 0 (resp. < 0), then x0 is a strict local minimum (resp.

maximum).

(b) If x0 is a local minimum (resp. maximum), then f ′′(x0) ≥ 0 (resp. ≤ 0).

(c) If f ′′(x0) > 0 (resp. < 0), there exists a neighborhood of x0 where for all x, f(x) ≥ g(x) (resp. ≤ g(x)).

(d) If f(x) ≥ g(x) (resp. ≤ g(x)) for all x in a neighborhood of x0, then f ′′(x0) ≥ 0 (resp. ≤ 0).

Proof. Proof of (a). If f ′′(x0) > 0, then for x close enough to x0, we write

f ′(x) = f ′′(x0)(x−x0) + o(x−x0),

so since f ′′(x0) > 0, there exists δ > 0 such that f ′(x) > 0 for x ∈ (x0,x0 + δ) and f ′(x) < 0 for

x ∈ (x0 − δ,x0). In particular, f is strictly decreasing on the left of x0 and strictly increasing on the right of x0, thus f has a local minimum at x0. The case f

′′(x0) is similar.

Proof of (b). By contradition, if x0 is local minimum and f ′′(x0) < 0, then by (a), x0 is also a

strict local maximum, which is impossible.

Proof of (c). Apply part (a) to the function f(x) −g(x). Proof of (d). Apply part (b) to the function f(x) −g(x).

Example 50. 1. In (b), it is not necessary to have f ′′(x0) > 0 at a strict local minimum: the function f(x) = x4 has a strict local minimum at x0 = 0, yet f

′′(0) = 0.

2. Moreover, when f ′′(x0) = 0, we cannot tell a priori if the graph is above or below the tangent line near x0, as this could be neither case, see for instance f(x) = x

3 at x = 0.

Convexity/concavity. Another way to look at the second derivative is when describing the position of a graph with respect to its local chords (lines joining two points on the curve).

MATH 105A - Fall 2018 - François Monard - UC Santa Cruz 93

y = f(x)

x0

y = f(x0) + f ′(x0)(x−x0)

x0

f ′′(x0) > 0 f ′′(x0) < 0

y = f(x)

Figure 13: Local considerations when f ′′(x0) 6= 0.

-4 -2 0 2 4

-3

-2

-1

0

1

2

3 y=x3

y=x4

y=-x4

Figure 14: Local considerations when f ′′(x0) = 0. Until we look at higher-order derivatives, no conclusion can be made regarding the relative position of f with respect to its tangent line near x0. Here, all three functions satisfy f(0) = f

′(0) = f ′′(0) = 0 so the tangent line at x = 0 is y = 0 but the relative positions can be anything near x = 0.

MATH 105A - Fall 2018 - François Monard - UC Santa Cruz 94

Theorem 86. Suppose f : [a,b] → R is of class C2 on (a,b). Let (x1,x2) ⊂ (a,b) and define17

g(x) := f(x1) + (x−x1) f(x2)−f(x1)

x2−x1 .

(i) If f ′′(x) > 0 for every x ∈ (x1,x2), then f(x) < g(x) for every x ∈ (x1,x2). (ii) If f ′′(x) < 0 for every x ∈ (x1,x2), then f(x) > g(x) for every x ∈ (x1,x2).

Proof. We only prove (i), as (ii) is similar. Set h = f − g. Note that h(x1) = h(x2) = 0 and h′′(x) = f ′′(x) > 0 for every x ∈ (x1,x2). Since h is continuous on [x1,x2] so achieves its maximum there. If there is x ∈ (x1,x2) such that h(x) ≥ 0, then h has a local maximum x0 inside (x1,x2), but from the previous theorem, h′′(x0) ≤ 0, which contradicts h′′ > 0.

The past two theorems tell us that if f ′′(x) > 0 throughout an interval, then the graph of f lies above its tangents and below its chords. More generally, we call a function f : [a,b] → R convex on [a,b] if for every x1 < x2 in [a,b], and t ∈ (0, 1),

f(tx1 + (1 − t)x2) ≤ tf(x1) + (1 − t)f(x2).

Similarly, f is concave on [a,b] if for every x1 < x2 in [a,b], and t ∈ (0, 1),

f(tx1 + (1 − t)x2) ≥ tf(x1) + (1 − t)f(x2). (20)

One may define strictly convex and strictly concave by making the inequalities strict in the last two equations above.

Noticing that in the last two right-hand sides,

tf(x1) + (1 − t)f(x2) = g(tx1 + (1 − t)x2),

where g(x) = f(x1) + (x−x1) f(x2)−f(x1)

x2−x1 , Theorem 86 is equivalent to saying that

(i) If f ′′(x) > 0 for every x ∈ [a,b], then f is strictly convex on [a,b]. (ii) If f ′′(x) < 0 for every x ∈ [a,b], then f is strictly concave on [a,b].

Example 51. 1. The function f(x) = x2 is strictly convex on R since f ′′(x) = 2 for all x.

2. The function f(x) = exp(x) is strictly convex on R since exp′′(x) = exp(x) > 0 for all x.

Graphically, a function f is convex on [a,b] if its epigraph epi(f) = {(x,y) : x ∈ [a,b],y ≥ f(x)} is a convex domain of R2 (in the sense that, if P1 = (x1,y1) and P2 = (x2,y2) belong to epi(f), then the segment P1P2 belongs entirely to epi(f)).

17g is the unique affine function passing through (x1,f(x1)) and (x2,f(x2)).

MATH 105A - Fall 2018 - François Monard - UC Santa Cruz 95

Exercises for Lecture 22:

1. Where are the following functions convex/concave ?

(a) f(x) = (x2 + 1) exp(x), x ∈ R. (b) f(x) =

√ x, x ∈ (0,∞).

2. Suppose that f : (a,b) → (c,d) is C2 and invertible (in particular, f ′(x) 6= 0 for every x ∈ (a,b)).

(a) Express (f−1)′′(y) solely in terms of f ′(f−1(y)) and f ′′(f−1(y)).

(b) If f is strictly convex on (a,b), is it always true that f−1 is strictly concave on (c,d) ? Prove or disprove.

3. (a) Prove that ln(x) is increasing and concave on (0,∞). (b) Prove by induction on n using (a) that for any positive numbers x1, . . . xn,

n √ x1 · · ·xn ≤

1

n (x1 + · · · + xn).

[Hint: this is equivalent to showing that 1 n

(ln(x1) + · · · + ln(xn)) ≤ ln ( 1 n

(x1 + . . .xn) ) .

Think about how to inductively use (20)]