If `a+b=20`, then the maximum value of ab is :

Options

A. 19
B. 75
C. 100
D. 120

Answer

To find the maximum value of \( ab \) given that \( a + b = 20 \), we can follow these steps:

### Step 1: Express \( b \) in terms of \( a \)
Since we know that \( a + b = 20 \), we can express \( b \) as:
\[ b = 20 - a \]

### Step 2: Substitute \( b \) in the expression for \( ab \)
Now, we can substitute \( b \) in the product \( ab \):
\[ ab = a(20 - a) = 20a - a^2 \]

### Step 3: Rewrite the expression
The expression \( ab = 20a - a^2 \) is a quadratic equation in terms of \( a \):
\[ ab = -a^2 + 20a \]

### Step 4: Find the vertex of the quadratic
The maximum value of a quadratic equation \( ax^2 + bx + c \) occurs at \( x = -\frac{b}{2a} \). Here, \( a = -1 \) and \( b = 20 \):
\[ a = -\frac{20}{2 \times -1} = 10 \]

### Step 5: Calculate \( b \)
Using the value of \( a \):
\[ b = 20 - a = 20 - 10 = 10 \]

### Step 6: Calculate the maximum value of \( ab \)
Now we can find the maximum value of \( ab \):
\[ ab = 10 \times 10 = 100 \]

Thus, the maximum value of \( ab \) when \( a + b = 20 \) is:
\[ \boxed{100} \]