HW 12
(1) Let a < b. Suppose f is differentiable on R. If f 0 (a) < y0 < f 0 (b), show that there exists
x0 ∈ (a, b) such that f 0 (x0 ) = y0 .
sin u
(2) Let f (t) = lim
, t ∈ R. Show that f is a continuous function on R. Define
u→t u
Z x
f (t)dt, x ∈ R.
S(x) =
0
(a) Find S(0), S 0 (0).
(b) Show that
Z
x
S(t)dt = xS(x) + cos x − 1.
0
(3) Let f (x) be a function on [0, 1] such that
(a) f ∈ C[0, 1], f (0) = 0, f (1) = 7.
(b) f 0 (x), f 00 (x) exist for 0 < x < 1.
(c) f 00 (x) > 0 for any 0 < x < 1.
Show that f (x) < 7x for any 0 < x < 1.
√
x
, x ≥ 0 and F 0 (x) = f (x). Prove that
(4) Let f (x) = 2
x +1
|F (x2 ) − F (y 2 )| ≤ |x − y|
for all x, y ≥ 0.
(5) Show that for x > 0,
1−
1
≤ ln x ≤ x − 1.
x
(6) Show that for all x > 0,
ln
1 2
x +x+1
2
< x.
(7) Suppose a < b. Show that
ea (b − a) < eb − ea < eb (b − a).
(8) Let f, g be differentiable functions on R such that
d
f (x) = −g(x)
dx
(9)
(10)
(11)
(12)
d
(xg(x)) = xf (x).
dx
(a) Show that between two consecutive roots of f (x) = 0, g(x) has a root.
(b) Show
two consecutive roots of g(x) = 0, f (x) has a root.
r that between
q
p
3
3
3
Let y = (x + 1) (x2 + 1) x3 + 1. Find y 0 (0).
x cos x
Let y =
, x ∈ R. Find y 0 (0).
(x + 1)(x + 2) · · · (x + n)
Let f (x) = (x2 + 1)sin x , x ∈ R. Find f 0 (1).
dy
Find
.
dx
3 cosh(x2 + 4)
(a) y = √
, x ∈ R.
x2 + 3x + 1
(b) y = sinh−1√
(ex ).
−1
(c) y = tan
x3 + 1, x ∈ R.
πx
(d) y = x , x > 0.
1
2
Z
x3
(e) y =
3
sin(t3 )e−t dt.
x2
2
ex
.
ln(3x4 + 5)
(13) Let a, b ∈ R. Show that the equation
(f) y =
2x3 − 3x2 − a2 x + b = 0
has at most one zero in [0, 1].
(14) Suppose that f is a nonzero continuous function on [a, b] differentiable on (a, b) and f (a) =
f (b) = 0. Show that for any λ ∈ R, there exists c ∈ (a, b) such that
f 0 (c) + λf (c) = 0.
Hint: consider h(x) = eλx f (x), x ∈ [a, b].
(15) Let f be a continuous function on R. Suppose that f satisfies
Z x
Z x
2x
f (t)dt = xe +
e−t f (t)dt, x ∈ R.
0
0
Find f (x).
(16) Let g : (−1, ∞) → R be a function such that
g(3x4 + 4x3 + 1) = ln(x + 2).
Find g 0 (8).
(17) Suppose that√f is a function on R such that f (tan x) = x for all x ∈ (−π/2, π/2). Find f 0 (1).
(18) Let f (x) = x3 + x + 6, x > 0? Is it true that f is an injection on (0, ∞)? If it is an
injection, let g be its inverse. Find g 0 (4).
(19) Let f (x) = 3 + x + x2 + tan(πx/2), −1 < x < 1.
(a) Show that f is increasing on (−1, 1).
0
(b) Let g be
Z the inverse function of f. Find g (3).
x
tesin t dt, x ∈ R. Show that F (x) ≥ 0 for all x ∈ R.
(20) Let F (x) =
0
(21) Suppose f ∈ C(−∞, ∞) and f (0) 6= 0. Suppose that
f (x + y) = f (x)f (y),
x, y ∈ R.
Show that f (0) > 0 and f (x) = (f (1))x for all x ∈ R.
(22) (a) Prove that the equation x3 +x2 +x = a has a unique real solution for every real number
a.
(b) Let x1 > 0. Define a sequence (xn ) of real numbers recursively by
x3n+1 + x2n+1 + xn+1 = xn ,
n ≥ 1.
Prove that (xn ) is convergent and find its limit.
(23) Let f : [a, b] → [c, d] be a continuous one-to-one and onto function.
(a) Show that f is either increasing or decreasing.
(b) Suppose a > 0, and f (x) > 0 on [a, b]. Let g : [c, d] → [a, b] be its inverse function,
Z b
g ∈ C[c, d]. Denote I =
f (x)dx. Using a, b, c, d and I to express
a
Z
d
g(y)dy.
c
(c) Let f : [0, 3] → [0, 3] be continuous and injective with f (0) = 0 and f (3) = 3. If
Z 3
Z 3
9
f (x)dx = , find
g(y)dy where g is the inverse function to f.
5
0
0
3
(d) Suppose further that f ∈ C 1 [a, b] and c = f (a) < f (b) = d and g ∈ C 1 [c, d] be its
inverse function as above. Show that
Z d
Z b
yg 0 (y)dy.
f (x)dx =
a
c
(24) Evaluate 1
1
(a) lim
−
.
x→1 ln x
x−1
!
Z x
1
1
π
√
(b) lim
dt −
.
x→1 x − 1
6
2t − t2
1/2
Z 1
√
1
3
t − 1 cos tdt.
(c) lim
2
x→1 (x − 1)
x
3x − 3sin x
(d) lim
.
x→0 x3
1
1
(e) lim
− x
.
x→0 x
e −1
Z 0
1
(f) lim 2
cos tdt.
x→0 3x
2
Z x3x
1
sin 2t
dt.
(g) lim
x→0+ x 0
t
x
(h) lim x .
x→0+
√
√
R x2
(sin t − t)dt
(i) lim R 0x2
√
√
x→0+
(tan t − t)dt
0
1/x2
sin x
(j) lim
.
x→0+
x
2x+1
2x − 3
(k) lim
.
x→∞ 2x + 5
x2 +1
2
x −4
.
(l) lim
x→∞ x2 − 1
x
−1
2 tan x
(m) lim
.
x→∞
πZ
x
1
ln tdt.
(n) lim
x→∞ x ln x 1
(o) lim x(e1/x − 1).
x→∞
Z x
1
1
−1
(p) lim
tan
dt
x→∞ ln x 1
t
1
(q) lim cosn .
n→∞
n
(25) Let y = f (x) be a function defined on a closed interval [a, b] such that f 0 , f 00 exist on (a, b)
and f is continuous on [a, b]. We say that f is concave up if
f (tx1 + (1 − t)x2 ) ≤ tf (x1 ) + (1 − t)f (x2 ),
t ∈ [0, 1]
for any x1 , x2 ∈ [a, b] and concave down if
f (tx1 + (1 − t)x2 ) ≥ tf (x1 ) + (1 − t)f (x2 ),
00
t ∈ [0, 1]
for any x1 , x2 ∈ [a, b]. We have proved that if f > 0 on (a, b) then f is concave up on (a, b);
if f 00 < 0, then f is concave up on (a, b).
(a) Let f (x) = ex for x ∈ R. Show that f is concave up on R.
(b) Let f (x) = ln x, for x > 0. Show that f is concave down on x > 0.
4
(c) Suppose that a, b > 0. Use (2) to show that
√
a+b
ab ≤
.
2
(d) Suppose λ1 , λ2 , λ3 ≥ 0 such that λ1 + λ2 + λ3 = 1. Assume that f is concave up on
(a, b). Show that for any x1 , x2 , x3 ∈ (a, b), we have
f (λ1 x1 + λ2 x2 + λ3 x3 ) ≤ λ1 f (x1 ) + λ2 f (x2 ) + λ3 f (x3 ).
(e) Assume that f is concave up on (a, b). Show that for any λ1 , · · · , λn ≥ 0 with
1 and any x1 , · · · , xn ∈ (a, b), one has
!
n
n
X
X
f
λ i xi ≤
λi f (xi ).
i=1
Pn
i=1
λi =
i=1
This inequality is called the Jensen inequality.
(f) Assume that a1 , · · · , an > 0. Use (2) to show that
√
a1 + a2 + · · · + an
n
a1 a2 · · · an ≤
.
n
(g) If A, B, C are angles of a triangle, show that
√
3 3
.
sin A + sin B + sin C ≤
2
(26) Show that 3x4 + 4x3 + 1 > 0 for all x ∈ R.
(27) Find the local extremum of the function
Z x
f (x) =
t(t − 1)2 (t + 1)3 dt, x ∈ R.
0
(28) Find the necessary and sufficient conditions on b, c such that the equation
x3 + 3bx + c = 0
has three distinct real roots. Here a, b ∈ R.
(29) Identify the intervals on which the following function are concave up and concave down,
decreasing and increasing. Also find all of its critical points, inflection points.
x4
(a) f (x) =
− 2x2 + 4 for x ∈ R.
4
(b) f (x) = x3 (10 −√3x2 ), x ∈ R.
3
(c) f (x) = (x − 5) x2 , x ∈ R.
(30) Let y = f (x) be a function defined on a subset D of R. (1) Determine the maximal possible
subset D ⊂ R where f (x) can be defined. (2) Find all of the vertical asymptotes of y = f (x).
(3) Find all of the oblique asymptotes of y = f (x). (4)Compute f 0 (x). (5) Find the critical
points of y = f (x). (6) Find the (local) maximum, maximum and the local minimum,
minimum of y = f (x). (7) Identify the intervals on which the function are increasing and
decreasing. (8) Compute f 00 (x). (9) Identify the intervals on which the function are concave
up and concave down. (10) Sketch the graph.
x
(a) y = ex−e .
(b) y = (x − 1)1/3 − 2(x − 1)4/3 .
(c) y = x2/3 (6 − x)1/3 .
x3
(d) y = 2
.
x −1
1
(e) y =
.
2x − 3
x 2
(f) y = +
2 x
x−4
(g) y =
x−5
5
x2 + x − 2
x−2
2
(i) y = x 3 e−x
1
(j) y = x2 + 2
x
(k) y 2 =p
x(x − 2),
(l) y = 1 − e−x2 .
8x
(m) y =
(x + 2)2 .
(31) Let F (x, y) = 6x2 + 3xy + 2y 2 + 17y − 6, (x, y) ∈ R2 . We call the set
(h) y =
C = {(x, y) : F (x, y) = 0}
the level curve of F. In this exercise, we are going to show that the level curve can be
identified with a function y = f (x) in a neighborhood of P (−1, 0).
(a) Show that F (−1, 0) = 0.
∂F
∂F
(b) Compute
and
∂x
∂y
∂F
(c) Show that
(P ) 6= 0.
∂y
(d) By (3) and using the implicit function theorem, we know that in a neighborhood of P,
the level curve defined a differentiable function y = f (x) so that 0 = f (−1). Compute
dy dx (x,y)=(−1,0)
(e) Find the equation of tangent line and of the normal line to C through P.
(32) Let E be the ellipse defined by the equation x2 + 2y 2 = 1. For each p ∈ E, denote sp by the
slope of the tangent line to E at p. Find
all p ∈ E such that sp = 1.
d3 y (33) Suppose xy + x − y + 1 − 0. Find
.
dx3 (x,y)=(2,−3)
(34) Suppose y = xx+y . Find y 0 (1).
(35) Find the equation of tangent line and of normal line to the curve C at the given point.
(a) C : x2 + xey + ln(y + 1) = 2 at (1, 0).
(b) C : −3x2 − 16xy − 2y 2 + 3y = 178 at (−3, 5).
dy
(36) Suppose cos(xy) = y 2 + 2x. Find
dx