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f:I->R; I is an interval on the real line; f is differentiable

(1) f is convex

(2) f' is increasing

Prove that (1) => (2)

(1) f is convex

(2) f' is increasing

Prove that (1) => (2)

Hint:

x<c<y from I, (f(c) - f(x)) / (c-x) <= (f(y) - f(c)) / (y-c)

x<c<y from I, (f(c) - f(x)) / (c-x) <= (f(y) - f(c)) / (y-c)

f:I->R; I is an interval on the real line; f is differentiable

(1) f is convex

(2) f' is increasing

Prove that (2) => (1)

(1) f is convex

(2) f' is increasing

Prove that (2) => (1)

Use Lagrange's mean-value theorem.

page 119 in palace

page 119 in palace

f, g, F : I -> R; I is an interval

F = max{f,g}

f, g are convex, does F have to be convex?

F = max{f,g}

f, g are convex, does F have to be convex?

Yes.

f, g, F : I -> R; I is an interval

F = min{f,g}

f, g are convex, does F have to be convex?

F = min{f,g}

f, g are convex, does F have to be convex?

No.

f(x)=7

g(x)=x

f(x)=7

g(x)=x

Prove Jensen's inequality for a convex function.

page 122 in palace

Prove that

(a1 * a2 * ... * an)^(1/n) <= (a1 + a2 + ... + an)/n

(a1 * a2 * ... * an)^(1/n) <= (a1 + a2 + ... + an)/n

Use Jensen's inequality for the minus logarithmic function,
which is convex.

Prove that

n / (1/a1 + 1/a2 + ... + 1/an) <= (a1 * a2 * ... * an)^(1/n)

n / (1/a1 + 1/a2 + ... + 1/an) <= (a1 * a2 * ... * an)^(1/n)

Use: (a1 * a2 * ... * an)^(1/n) <= (a1 + a2 + ... + an)/n

(1) f:(0;oo)->|R

(2) f is convex

(3) f is strictly increasing

Prove that lim f(x) = infinity, as x approaches infinity.

(2) f is convex

(3) f is strictly increasing

Prove that lim f(x) = infinity, as x approaches infinity.

Choose 0<a<b. Put k = (f(b) - f(a)) / (b-a). By (3) k>0.

Since f is convex conclude that

for all x > b, kx + (f(b)-kb) <= f(x).

And hence conclude the thesis.

Since f is convex conclude that

for all x > b, kx + (f(b)-kb) <= f(x).

And hence conclude the thesis.

f,g : I -> R; I is an interval on the real line;

f,g are convex; a,b are positive numbers

Prove that af+bg is convex.

f,g are convex; a,b are positive numbers

Prove that af+bg is convex.

page 124 in palace

(1) f : [0,oo) -> |R

(2) f is convex

(3) f(0) = 0

(4) g : (0,oo) -> |R , g(x)=f(x)/x

What can we conclude about function g?

(2) f is convex

(3) f(0) = 0

(4) g : (0,oo) -> |R , g(x)=f(x)/x

What can we conclude about function g?

g is increasing

Hint: If f is convex, then for all x<c<y

(f(c) - f(x)) / (c-x) <= (f(y) - f(c)) / (y-c).

Use f(0)=0.

page 124 in the palace notebook

Hint: If f is convex, then for all x<c<y

(f(c) - f(x)) / (c-x) <= (f(y) - f(c)) / (y-c).

Use f(0)=0.

page 124 in the palace notebook