Question

If p and q are positive real numbers such that ${p^2} + {q^2} = 1$, then the maximum value of $\left( {p + q} \right)$ is
1. $2$
2. $\dfrac{1}{2}$
3. $\dfrac{1}{{\sqrt 2 }}$
4. $\sqrt 2$

Answer 1

we are given two positive real numbers and we have to find the maximum value of their sum. For this we will use the concept of Arithmetic Mean and Geometric Mean of numbers. Consider, if ${a_1},{a_2},...,{a_n}$ are the observations, then the arithmetic mean is defined as
$AM = \dfrac{{{a_1},{a_2},...,{a_n}}}{n}$ and the geometric mean is defined as $GM = {\left( {{a_1},{a_2},...,{a_n}} \right)^{\dfrac{1}{n}}}$ . Also we will use the relation between them as $AM \geqslant GM$.

Complete step-by-step solution:
Arithmetic mean represents a number that is obtained by dividing the sum of the elements of a set by the number of values in the set.
Consider, if ${a_1},{a_2},...,{a_n}$ are the observations, then the A.M. is defined as
$AM = \dfrac{{{a_1},{a_2},...,{a_n}}}{n}$
The Geometric Mean (GM) is the average value or mean which signifies the central tendency of the set of numbers by finding the product of their values.
In other words, the geometric mean is defined as the nth root of the product of n numbers
Consider, if ${a_1},{a_2},...,{a_n}$ are the observations, then the G.M is defined as
$GM = {\left( {{a_1},{a_2},...,{a_n}} \right)^{\dfrac{1}{n}}}$
Relation between arithmetic mean and geometric mean is as follows:
$AM \geqslant GM$

We know that $AM \geqslant GM$
Applying this property to ${p^2}$ and ${q^2}$ we get ,
$\dfrac{{{p^2} + {q^2}}}{2} \geqslant pq$
Since we are given that ${p^2} + {q^2} = 1$. Therefore we get ,
$\dfrac{1}{2} \geqslant pq$
Therefore $1 \geqslant 2pq$
Now consider
${\left( {p + q} \right)^2} = {p^2} + {q^2} + 2pq$
$= 1 + 2pq$
Here for the maximum value of ${\left( {p + q} \right)^2}$ the value of pq should be maximum which is 1 according to the inequality $1 \geqslant 2pq$.
$= 1 + 1 = 2$
Therefore we get $\left( {p + q} \right) = \sqrt 2$
Hence we get the required value.
Therefore option (4) is the correct answer.

Note: in order to solve such type of questions one must be well versed with the concept of Arithmetic mean (AM) and Geometric Mean (GM) .Keep in mind that the geometric mean is different from the arithmetic mean. Remember the relation between Arithmetic mean (AM) and Geometric Mean (GM) as $AM \geqslant GM$ .