Question

If the function $f(x) = x^2 + ax + 1$ is increasing on $[1,2]$, then $a$ is greater than or equal to

• A. $-2$
• B. $-5$
• C. $-4$
• D. $-7$
• E. $-3$

Hint:

For increasing functions: - Check $f'(x) \geq 0$ - For linear derivatives, minimum occurs at the left endpoint of the interval.

Answer 1

The Correct Option is A

Concept: A function is increasing on an interval if its derivative is non-negative throughout that interval: \[ f'(x) \geq 0 \]

Step 1: Find the derivative of the given function.
\[ f(x) = x^2 + ax + 1 \] \[ f'(x) = 2x + a \]

Step 2: Apply the condition for increasing function on $[1,2]$.
For $f(x)$ to be increasing on $[1,2]$, we need: \[ 2x + a \geq 0 \quad \text{for all } x \in [1,2] \]

Step 3: Find the minimum value of $2x + a$ on $[1,2]$.
Since $2x + a$ is increasing in $x$, its minimum occurs at $x = 1$: \[ 2(1) + a \geq 0 \]

Step 4: Solve the inequality.
\[ 2 + a \geq 0 \quad \Rightarrow \quad a \geq -2 \]