Question

Assertion :If $3x+4y=5$ then greatest value of $x^2{y}^{3}is\frac{3}{16}$ Reason: Greatest value occurs when $9x=8y.$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$3x+4y=5i.e.ax+by=k$

the expression $x^m{y}^n(m,n\ge 1)$ will be maximum when ${\left(\frac{ax}{m}\right)}^m\times {\left(\frac{by}{n}\right)}^n$ is maximum & this will be maximum at

$\frac{ax}{m}=\frac{by}{n}=\frac{ax+by}{m+n}=\frac{k}{m+n}$

$(Herea=3,b=4,m=2,n=3,k=5)$

$\therefore \frac{3x}{2}=\frac{4y}{3}=\frac{5}{5}$

$\Rightarrow x=\frac{2}{3},y=\frac{3}{4}\Rightarrow \frac{x}{y}=\frac{8}{9}or9x=8y$

$\therefore$ Maximum value of $x^2{y}^3={\left(\frac{2}{3}\right)}^2{\left(\frac{3}{4}\right)}^3=\frac{3}{16}$

$\therefore$ Assertion (A) & Reason (R) both are true & Reason (R) is the correct explanation of Assertion (A).