Question

If $a+b=10$ and $ab$ is maximum, then the value of a is

• A. 5
• B. 3
• C. 6
• D. 25
• E. 10

Hint:

A square has the largest area among all rectangles with the same perimeter.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We are asked to find the value of 'a' that maximizes the product 'ab' given a fixed sum 'a+b'. This is a classic optimization problem that can be solved using calculus or the AM-GM inequality.
Step 2: Key Formula or Approach:
Method 1: Calculus
1. Express the product as a function of a single variable.
2. Find the derivative of this function and set it to zero to find critical points.
3. Use the second derivative test to confirm it's a maximum.
Method 2: AM-GM Inequality
For non-negative numbers, the Arithmetic Mean (AM) is always greater than or equal to the Geometric Mean (GM).
\[ \frac{a+b}{2} \geq \sqrt{ab} \] Equality holds when \( a = b \).
Step 3: Detailed Explanation:
Using Method 1 (Calculus):
Let the product be \( P = ab \).
From the given condition, \( a+b=10 \), we can write \( b = 10-a \).
Substitute this into the product equation to get a function of 'a':
\[ P(a) = a(10-a) = 10a - a^2 \] To find the maximum, we find the derivative with respect to 'a' and set it to 0.
\[ \frac{dP}{da} = 10 - 2a \] Set the derivative to zero:
\[ 10 - 2a = 0 \implies 2a = 10 \implies a = 5 \] To confirm it's a maximum, we check the second derivative:
\[ \frac{d^2P}{da^2} = -2 \] Since the second derivative is negative, the function has a maximum at \( a=5 \).
Using Method 2 (AM-GM Inequality):
We have \( a+b=10 \). Assuming a and b are positive (which they must be for the product to be maximized in this context), we can apply the AM-GM inequality:
\[ \frac{a+b}{2} \geq \sqrt{ab} \] \[ \frac{10}{2} \geq \sqrt{ab} \] \[ 5 \geq \sqrt{ab} \] Squaring both sides:
\[ 25 \geq ab \] The maximum value of the product ab is 25. This maximum is achieved when the equality holds in the AM-GM inequality, which happens when \( a = b \).
Since \( a+b=10 \) and \( a=b \), we have \( a+a=10 \implies 2a=10 \implies a=5 \).
Step 4: Final Answer:
The value of a is 5.