3 numbers x,y,z are in G.P,x,y2,z are in A.P., then minimum value of x+2y+z is equal to

Answers (1)

x, y, z are in G.P.

y^2 = xz    -(i)

x, y^2, z are in A.P.

y^2 = \frac{x+z}{2}    -(ii)

xz = \frac{x+z}{2}

2xz = x+z

Now, A = x+ 2y+ z

\\ = 2y^2 + 2y         -[From (i) and (ii)]

\\ \frac{dA}{dy} = 4y+2 = 0 \\ \Rightarrow y = -\frac{1}{2}

\\ \frac{d^2A}{dy^2} = 4 > 0

\therefore

 \\ A_{min}\ for\ y = -\frac{1}{2} \\ A_{min} = 2(-\frac{1}{2})^2 + 2(-\frac{1}{2}) \\ = 2(\frac{1}{4})-1 \\ = -\frac{1}{2}

 

Most Viewed Questions

Preparation Products

JEE Foundation + Knockout JEE Main 2025 (Easy Installment)

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 3999/-
Buy Now
JEE Foundation + Knockout JEE Main 2024 (Easy Installment)

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 3999/-
Buy Now
JEE Foundation 2023 (Easy Installment)

Adaptive Time Table, 450 hours of Video Lectures, NCERT Solutions, CBSE Previous Year Papers, Chapter wise Tests,.

₹ 4999/-
Buy Now
JEE Main Rank Booster 2022

Booster and Kadha Video Lectures, Unlimited Full Mock Test, Adaptive Time Table, 24x7 Doubt Chat Support,.

₹ 13999/- ₹ 9999/-
Buy Now
Knockout BITSAT 2021

It is an exhaustive preparation module made exclusively for cracking BITSAT..

₹ 4999/- ₹ 2999/-
Buy Now
Boost your Preparation for JEE Main 2021 with Personlized Coaching
 
Exams
Articles
Questions