Question

If $f(x) = x^2 + bx + 1$ is increasing in the interval $[1, 2]$, then the least value of $b$ is:

• A. 5
• B. 0
• C. -2
• D. -4

Hint:

When analyzing the behavior of a function, especially in terms of increasing or decreasing, it’s important to check the derivative at the boundary points of the given interval. In this case, by ensuring \( f'(x) \geq 0 \) at the endpoints \( x = 1 \) and \( x = 2 \), we can determine the least value of \( b \) that ensures the function is increasing on the entire interval. Always remember to compare the conditions at all boundary points to find the solution.

Answer 1

The Correct Option is C

The function \( f(x) = x^2 + bx + 1 \) is increasing if \( f'(x) \geq 0 \) for all \( x \in [1, 2] \). Differentiating \( f(x) \):

\[ f'(x) = 2x + b. \]

For \( f'(x) \geq 0 \) in \([1, 2]\), check the boundary points:

At \( x = 1 \):

\[ 2(1) + b \geq 0 \implies b \geq -2. \]

At \( x = 2 \):

\[ 2(2) + b \geq 0 \implies b \geq -4. \]

Thus, the least \( b \) satisfying both conditions is \( b = -2 \).

Answer 2

The function \( f(x) = x^2 + bx + 1 \) is increasing if \( f'(x) \geq 0 \) for all \( x \in [1, 2] \). Let’s find the value of \( b \) for which this condition is satisfied.

Step 1: Differentiate the function \( f(x) \):

\[ f'(x) = 2x + b \]

Step 2: Analyze the condition for \( f'(x) \geq 0 \) in \([1, 2]\):

For the function to be increasing on the interval \([1, 2]\), the derivative \( f'(x) \) must be non-negative for all \( x \in [1, 2] \). Therefore, we need to check the boundary points \( x = 1 \) and \( x = 2 \), and ensure that the derivative is greater than or equal to zero at both points.

Step 3: Check the boundary point at \( x = 1 \):

Substituting \( x = 1 \) into the derivative equation: \[ 2(1) + b \geq 0 \implies b \geq -2 \]

Step 4: Check the boundary point at \( x = 2 \):

Substituting \( x = 2 \) into the derivative equation: \[ 2(2) + b \geq 0 \implies b \geq -4 \]

Step 5: Determine the least value of \( b \):

From the two conditions, we have \( b \geq -2 \) from \( x = 1 \), and \( b \geq -4 \) from \( x = 2 \). The least value of \( b \) that satisfies both conditions is \( b = -2 \).

Conclusion: Thus, the least value of \( b \) that makes \( f(x) \) increasing on the interval \([1, 2]\) is \( b = -2 \).