A question regarding minimum value of...

Question

$a+b+c=2$

$a,b,c \in \mathbb{R}^+$

What is The min value of $1/a+4/b+9/c$?

I have tried to use geometric shapes and aritmetic harmonic geometric means but unfortunately ı am going nowhere?

What do you suggest?

Answer 1

Lagrange multiplier

$$f(a,b,c,k)=\frac{1}{a}+\frac{4}{b}+\frac{9}{c}+k (a+b+c-2)$$

Solve the system $f'_x=0,f'_y=0,f'_z=0,f'_k=0$

That is

$$k-\frac{1}{a^2}=0,k-\frac{4}{b^2}=0,k-\frac{9}{c^2}=0,a+b+c-2=0$$

there are $3$ solutions but only one leads to the minimum and respects constraints

$a= \frac{1}{3},b=\frac{2}{3},c= 1$ for which $\frac{1}{a}+\frac{4}{b}+\frac{9}{c}=18$

Hope this helps

Answer 2

Hint

Using AM-GM:

$$\frac{a+b+c}{3}\ge\sqrt[3]{abc}$$

and

$$\frac{1}{a}+\frac{4}{b}+\frac{9}{c}\ge3\sqrt[3]{\frac{36}{abc}}$$

Answer 3

Use Cauchy Schwarz inequality $ (\frac{1}{a} + \frac{4}{b} + \frac{9}{c}) (a + b + c) \ge (1+2+3)^2$.