If a, b, c are three positive numbers such th | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) By the Arithmetic Mean-Geometric Mean $(AM-GM)$ inequality,for positive numbers $a, b, c/2, c/2$:$\frac{a + b + c/2 + c/2}{4} \geq \sqrt[4]{a \cdo

Vedclass · Accepted Answer

$a = b = 1/4, c = 1/2$

Vedclass · Answer

(B) By the Arithmetic Mean-Geometric Mean $(AM-GM)$ inequality,for positive numbers $a, b, c/2, c/2$:$\frac{a + b + c/2 + c/2}{4} \geq \sqrt[4]{a \cdot b \cdot \frac{c}{2} \cdot \frac{c}{2}}$$\frac{a + b + c}{4} \geq \sqrt[4]{\frac{abc^2}{4}}$Raising both sides to the power of $4$:$\frac{(a + b + c)^4}{256} \geq \frac{abc^2}{4}$$abc^2 \leq \frac{(a + b + c)^4}{64}$Given the maximum value of $abc^2$ is $1/64$,we have $\frac{(a + b + c)^4}{64} = \frac{1}{64}$,which implies $a + b + c = 1$.The equality holds when $a = b = c/2$.Substituting $a = b = c/2$ into $a + b + c = 1$:$c/2 + c/2 + c = 1 \implies 2c = 1 \implies c = 1/2$.Then $a = b = 1/4$.

If $a, b, c$ are three positive numbers such that the maximum value of $abc^2$ is $1/64$,then:

Similar Questions

Let $a, b, c$ be the lengths of three sides of a triangle satisfying the condition $(a^2+b^2)x^2-2b(a+c)x+(b^2+c^2)=0$. If the set of all possible values of $x$ is the interval $(\alpha, \beta)$,then $12(\alpha^2+\beta^2)$ is equal to.

What is the minimum value of $\frac{1 - x + x^2}{1 + x + x^2}$?

The least value of $\frac{x^2y^2 - 2x^2y + 2x^2 + 2xy - 2x + 1}{x^2y + x}$ is $\lambda$,where $x, y \in R^+$ and $x^2y + x \neq 0$. Then:

Let $[r]$ denote the greatest integer not exceeding $r$. The roots of the equation $3 x^2 + 6 x + 5 + \alpha (x^2 + 2 x + 2) = 0$ are complex numbers whenever $\alpha > L$ or $\alpha < M$. If $(L - M)$ is minimum,then the greatest value of $[r]$ such that $L y^2 + M y + r < 0$ for all $y \in R$ is:

If $x$ is real, then the value of $\frac{x^2-3x+4}{x^2+3x+4}$ lies in the interval

Vedclass Test Series

Exam Paper Generator

Online Exam Module