Show that h(x

HW 11

(1) Let f be a twice differentiable function on R such that f⁰⁰(x) + f (x) = 0, for all x ∈ R. Suppose f (0) = 1 and f⁰(0) = 0.

(a) Define h(x) = (f⁰(x))²+ (f (x))², x ∈ R. Show that h(x) = 1 for all x ∈ R.

(b) Define g(x) = (f⁰(x) + sin x)²+ (f (x) − cos x)², x ∈ R. Show that g(x) = 0 for all x ∈ R.

(c) Conclude that f (x) = cos x for all x ∈ R.

(2) Let f (x) be a twice differentiable function on R and f⁰⁰(x) = 0. Show that there exist a, b ∈ R such that f (x) = ax + b. In fact, a = f⁰(0) and b = f (0). In general, suppose that f is (n + 1)-times differentiable on R and f⁽ⁿ⁺¹⁾(x) = 0 for all x ∈ R. Show that f is a polynomial function on R of degree at most n, i.e.

f (x) = a_nxⁿ+ a_n−1xⁿ⁻¹+ · · · + a₁x + a₀, x ∈ R, for some a₀, · · · , a_n ∈ R

(3) Let a_ij(x) be differentiable functions on an open interval I of R for 1 ≤ i, j ≤ 3. Let

f (x) =      

a11(x) a12(x) a13(x) a21(x) a22(x) a23(x) a31(x) a32(x) a33(x)

, x ∈ R.

Then f (x) is differentiable on I. Show that

f⁰(x) =      

a⁰₁₁(x) a⁰₁₂(x) a⁰₁₃(x) a21(x) a22(x) a23(x) a31(x) a32(x) a33(x)

+      

a11(x) a12(x) a13(x) a⁰₂₁(x) a⁰₂₂(x) a⁰₂₃(x) a31(x) a32(x) a33(x)

+      

a11(x) a12(x) a13(x) a21(x) a22(x) a23(x) a⁰₃₁(x) a⁰₃₂(x) a⁰₃₃(x)       (4) Let a₁, a₂, a₃∈ R \ {0}. Show that

1 + a₁ 1 1

1 1 + a₂ 1

1 1 1 + a₃

= a₁a₂a₃

 1 + 1

a1

+ 1 a2

+ 1 a3



using following steps. Define f (x) =

x + a1 x x

x x + a2 x

x x x + a3

, x ∈ R.

(a) Evaluate f (0) and f⁰(0).

(b) Show that f⁰⁰(x) = 0. Hence f (x) = f⁰(0)x + f (0).

(c) Find f (x) and evaluate f (1).

(5) Let f ∈ C²[a, b]. Suppose f (a) = f (b) = 0 and f⁰⁰(x) < 0 for all a < x < b. Show that f (x) > 0 for all a < x < b.

(6) Let f ∈ C¹[a, b] and f (a) = 0. Assume that M = sup

x∈[a,b]

|f (x)|.

(a) Show that there exists x0∈ [a, b] such that M = |f (x0)|.

(b) Show that Z b

a

|f⁰(x)|dx ≥ M.

(c) Show that M²≤ (b − a) Z b

a

|f⁰(x)|²dx.

(7) Define a function f (x) on [0, 1] by f (x) = x

2(x + 1), x ≥ 0.

(a) Find f⁰(x) and show that f is increasing on [0, 1].

1

2

(b) Let f ([0, 1]) be the set of all f (x) with 0 ≤ x ≤ 1. Show that f ([0, 1]) ⊂ [0, 1].

(c) Let a1= 1. Define

an+1= f (an), n ≥ 1.

Show that (an) is convergent and find its limit (8) For x > 2, show that

3

2 > (x + 1) cos π

x + 1− x cosπ x> 1.

(9) Let f ∈ C(−∞, ∞). We say that f is periodic of period 2π if f (x + 2π) = f (x) for all x ∈ R.

Show that for any a ∈ R,

Z a+2π a

f (x)dx = Z 2π

0

f (x)dx.

Hint: define a function F (a) = Z a+2π

a

f (x)dx, −∞ < a < ∞. By fundamental theorem of calculus, F ∈ C¹(−∞, ∞). Show that F is a constant function.

(10) Let f ∈ C(−∞, ∞).

(a) We say that f is an odd function if f (−x) = −f (x). Show that for any a ∈ R, Z a

−a

f (x)dx = 0 if f is an odd function.

(b) We say that f is an even function if f (−x) = f (x). Show that for any a ∈ R, Z a

−a

f (x)dx = 2 Z a

0

f (x)dx.

(11) Evaluate Z π

−π

 sin²⁰¹⁵x

4 + cos²x+ cos²x sin²x

 dx.

(12) Let f ∈ C[0, 1]. Prove the following equalities.

(a) Z 1

0

f (x)dx = Z 1

0

f (1 − x)dx.

(b) Z π

0

xf (sin x)dx = π 2

Z π 0

f (sin x)dx.

(c) Z ^π₂

0

f (sin 2x) cos xdx = Z ^π₄

0

f (sin 2x) cos xdx + Z ^π₄

0

f (sin 2x) sin xdx. Hint: Write Z ^π₂

0

· · · dx = Z ^π₄

0

· · · dx + Z ^π₂

π 4

· · · dx.

(13) Let n be a natural number.

(a) Let

I_n= Z ^π₂

0

sinⁿxdx.

Using sinⁿx = sinⁿ⁻¹x sin x and the integration by parts formula to write down the relation between In and In−2.

(b) Let

Jn= Z ^π₂

0

cosⁿxdx.

Similarly, write down the relation between J_n and J_n−2.