HW 12
(1) Let a < b. Suppose f is differentiable on R. If f0(a) < y0 < f0(b), show that there exists x0∈ (a, b) such that f0(x0) = y0.
(2) Let f (t) = lim
u→t
sin u
u , t ∈ R. Show that f is a continuous function on R. Define S(x) =
Z x 0
f (t)dt, x ∈ R.
(a) Find S(0), S0(0).
(b) Show that
Z x 0
S(t)dt = xS(x) + cos x − 1.
(3) Let f (x) be a function on [0, 1] such that (a) f ∈ C[0, 1], f (0) = 0, f (1) = 7.
(b) f0(x), f00(x) exist for 0 < x < 1.
(c) f00(x) > 0 for any 0 < x < 1.
Show that f (x) < 7x for any 0 < x < 1.
(4) Let f (x) =
√x
x2+ 1, x ≥ 0 and F0(x) = f (x). Prove that
|F (x2) − F (y2)| ≤ |x − y|
for all x, y ≥ 0.
(5) Show that for x > 0,
1 − 1
x≤ ln x ≤ x − 1.
(6) Show that for all x > 0,
ln 1
2x2+ x + 1
< 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
dxf (x) = −g(x) d
dx(xg(x)) = xf (x).
(a) Show that between two consecutive roots of f (x) = 0, g(x) has a root.
(b) Show that between two consecutive roots of g(x) = 0, f (x) has a root.
(9) Let y = 3 r
(x + 1)3 q
(x2+ 1)p3
x3+ 1. Find y0(0).
(10) Let y = x cos x
(x + 1)(x + 2) · · · (x + n), x ∈ R. Find y0(0).
(11) Let f (x) = (x2+ 1)sin x, x ∈ R. Find f0(1).
(12) Find dy dx.
(a) y = 3 cosh(x2+ 4)
√x2+ 3x + 1, x ∈ R.
(b) y = sinh−1(ex).
(c) y = tan−1√
x3+ 1, x ∈ R.
(d) y = xπx, x > 0.
1
(e) y = Z x3
x2
sin(t3)e−t3dt.
(f) y = ex2 ln(3x4+ 5).
(13) Let a, b ∈ R. Show that the equation
2x3− 3x2− a2x + 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
f0(c) + λf (c) = 0.
Hint: consider h(x) = eλxf (x), x ∈ [a, b].
(15) Let f be a continuous function on R. Suppose that f satisfies Z x
0
f (t)dt = xe2x+ Z x
0
e−tf (t)dt, x ∈ R.
Find f (x).
(16) Let g : (−1, ∞) → R be a function such that
g(3x4+ 4x3+ 1) = ln(x + 2).
Find g0(8).
(17) Suppose that f is a function on R such that f (tan x) = x for all x ∈ (−π/2, π/2). Find f0(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 g0(4).
(19) Let f (x) = 3 + x + x2+ tan(πx/2), −1 < x < 1.
(a) Show that f is increasing on (−1, 1).
(b) Let g be the inverse function of f. Find g0(3).
(20) Let F (x) = Z x
0
tesin tdt, x ∈ R. Show that F (x) ≥ 0 for all x ∈ R.
(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))xfor 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, g ∈ C[c, d]. Denote I =
Z b a
f (x)dx. Using a, b, c, d and I to express Z d
c
g(y)dy.
(c) Let f : [0, 3] → [0, 3] be continuous and injective with f (0) = 0 and f (3) = 3. If Z 3
0
f (x)dx = 9 5, find
Z 3 0
g(y)dy where g is the inverse function to f.
(d) Suppose further that f ∈ C1[a, b] and c = f (a) < f (b) = d and g ∈ C1[c, d] be its inverse function as above. Show that
Z b a
f (x)dx = Z d
c
yg0(y)dy.
(24) Evaluate (a) lim
x→1
1
ln x − 1 x − 1
. (b) lim
x→1
1 x − 1
Z x 1/2
√ 1
2t − t2dt −π 6
! .
(c) lim
x→1
1 (x − 1)2
Z 1 x
√3
t − 1 cos tdt.
(d) lim
x→0
3x− 3sin x x3 . (e) lim
x→0
1
x− 1
ex− 1
. (f) lim
x→0
1 3x2
Z 0 x2
cos tdt.
(g) lim
x→0+
1 x
Z 3x 0
sin 2t t dt.
(h) lim
x→0+xx. (i) lim
x→0+
Rx2 0 (sin√
t −√ t)dt Rx2
0 (tan√ t −√
t)dt (j) lim
x→0+
sin x x
1/x2
. (k) lim
x→∞
2x − 3 2x + 5
2x+1 .
(l) lim
x→∞
x2− 4 x2− 1
x2+1 . (m) lim
x→∞
2 tan−1x π
x . (n) lim
x→∞
1 x ln x
Z x 1
ln tdt.
(o) lim
x→∞x(e1/x− 1).
(p) lim
x→∞
1 ln x
Z x 1
tan−1 1 t
dt (q) lim
n→∞cosn 1 n.
(25) Let y = f (x) be a function defined on a closed interval [a, b] such that f0, f00 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), t ∈ [0, 1]
for any x1, x2∈ [a, b]. We have proved that if f00> 0 on (a, b) then f is concave up on (a, b);
if f00< 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.
(c) Suppose that a, b > 0. Use (2) to show that
√
ab ≤ a + b 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 (λ1x1+ λ2x2+ λ3x3) ≤ λ1f (x1) + λ2f (x2) + λ3f (x3).
(e) Assume that f is concave up on (a, b). Show that for any λ1, · · · , λn ≥ 0 withPn i=1λi= 1 and any x1, · · · , xn∈ (a, b), one has
f
n
X
i=1
λixi
!
≤
n
X
i=1
λif (xi).
This inequality is called the Jensen inequality.
(f) Assume that a1, · · · , an > 0. Use (2) to show that
√n
a1a2· · · an≤a1+ a2+ · · · + an
n .
(g) If A, B, C are angles of a triangle, show that sin A + sin B + sin C ≤ 3√
3 2 . (26) Show that 3x4+ 4x3+ 1 > 0 for all x ∈ R.
(27) Find the local extremum of the function f (x) =
Z x 0
t(t − 1)2(t + 1)3dt, x ∈ R.
(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.
(a) f (x) = x4
4 − 2x2+ 4 for x ∈ R.
(b) f (x) = x3(10 − 3x2), x ∈ R.
(c) f (x) = (x − 5)√3
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 f0(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 f00(x). (9) Identify the intervals on which the function are concave up and concave down. (10) Sketch the graph.
(a) y = ex−ex.
(b) y = (x − 1)1/3− 2(x − 1)4/3. (c) y = x2/3(6 − x)1/3.
(d) y = x3 x2− 1. (e) y = 1
2x − 3. (f) y = x
2 +2 x (g) y = x − 4
x − 5
(h) y = x2+ x − 2 x − 2 (i) y = x23e−x (j) y = x2+ 1
x2 (k) y2= x(x − 2),
(l) y =p
1 − e−x2. (m) y = 8x
(x + 2)2.
(31) Let F (x, y) = 6x2+ 3xy + 2y2+ 17y − 6, (x, y) ∈ R2. We call the set 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.
(b) Compute ∂F
∂x and ∂F
∂y (c) Show that ∂F
∂y(P ) 6= 0.
(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+ 2y2= 1. For each p ∈ E , denote spby the slope of the tangent line to E at p. Find all p ∈ E such that sp= 1.
(33) Suppose xy + x − y + 1 − 0. Find d3y dx3
(x,y)=(2,−3)
. (34) Suppose y = xx+y. Find y0(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 − 2y2+ 3y = 178 at (−3, 5).
(36) Suppose cos(xy) = y2+ 2x. Find dy dx
