If `a,b,c,d ,e` are +ve real numbers such that `a+b+c+d+e=8 and a^2 + b^2 +c^2 + d^2 +e^2 = 16`, then the range of 'e' is

Answer

To find the range of \( e \) given the conditions \( a + b + c + d + e = 8 \) and \( a^2 + b^2 + c^2 + d^2 + e^2 = 16 \), we can follow these steps:

### Step 1: Express \( a + b + c + d \) and \( a^2 + b^2 + c^2 + d^2 \) in terms of \( e \)

From the first equation, we can express \( a + b + c + d \) as:
\[ a + b + c + d = 8 - e \tag{1} \]

From the second equation, we can express \( a^2 + b^2 + c^2 + d^2 \) as:
\[ a^2 + b^2 + c^2 + d^2 = 16 - e^2 \tag{2} \]

### Step 2: Apply the Cauchy-Schwarz inequality

According to the Cauchy-Schwarz inequality:
\[ \frac{(a + b + c + d)^2}{4} \leq \frac{(a^2 + b^2 + c^2 + d^2)}{4} \]

Substituting equations (1) and (2) into the inequality gives:
\[ \frac{(8 - e)^2}{4} \leq \frac{(16 - e^2)}{4} \]

### Step 3: Simplify the inequality

Multiplying both sides by 4 (since 4 is positive):
\[ (8 - e)^2 \leq 16 - e^2 \]

Expanding the left side:
\[ 64 - 16e + e^2 \leq 16 - e^2 \]

### Step 4: Rearranging the inequality

Combining like terms:
\[ 64 - 16e + e^2 + e^2 - 16 \leq 0 \]
\[ 2e^2 - 16e + 48 \leq 0 \]

### Step 5: Factor the quadratic inequality

Dividing the entire inequality by 2:
\[ e^2 - 8e + 24 \leq 0 \]

Now we need to find the roots of the quadratic equation \( e^2 - 8e + 24 = 0 \) using the quadratic formula:
\[ e = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -8, c = 24 \):
\[ e = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot 24}}{2 \cdot 1} \]
\[ e = \frac{8 \pm \sqrt{64 - 96}}{2} \]
\[ e = \frac{8 \pm \sqrt{-32}}{2} \]

Since the discriminant is negative, the quadratic does not have real roots, which means it does not cross the x-axis.

### Step 6: Determine the sign of the quadratic

Since the leading coefficient (of \( e^2 \)) is positive, the quadratic \( e^2 - 8e + 24 \) is always positive. Therefore, we need to find the range of \( e \) that satisfies the original conditions.

### Step 7: Analyze the constraints on \( e \)

Since \( a, b, c, d, e \) are positive real numbers, we have:
1. \( e > 0 \)
2. From \( a + b + c + d + e = 8 \), it follows that \( e < 8 \).

### Step 8: Combine the results

Thus, the range of \( e \) is:
\[ 0 < e < 8 \]

### Final Result

The range of \( e \) is:
\[ (0, 8) \]

---