Question

Assertion :${(a+b)}^{1/n}\le a^{1/n}+b^{1/n},$ where $a,b\ge 0$ and $n\ge 1$ Reason: The function $f\left(x\right)={(1+x)}^p-x^p-1,x\ge 0$ and $0<p\le 1$ decreases in $(0,\infty )$

• 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

We have,

$f\left(x\right)={(1+x)}^p-x^p-1,x\ge 0$ and $0<p\le 1$

Then, $f\left(x\right)=p{(1+x)}^{p-1}-{px}^{p-1}$

$=p[\frac{1}{{(1+x)}^{1-p}}-\frac{1}{x^{1-p}}]<0\forall x>0$

$\Rightarrow f\left(x\right)$ strictly decreases in $(0,\infty )$

$\Rightarrow f\left(x\right)\le f\left(0\right)=0$

i.e., ${(1+x)}^p\le 1+x^p\forall x\ge 0$ and $0<p\le 1$

Now, putting $x=\frac{a}{b}$ and $p=\frac{1}{n}(n\ge 1),$ we get

${(\frac{a}{b}+1)}^{1/n}\le {\left(\frac{a}{b}\right)}^{1/n}+1$

$\Rightarrow {(a+b)}^{1/n}\le a^{1/n}+b^{1/n}$