Question 32 of 82intermediate🔧 ApplyShort Answer4 marks

For what values of a the function f given by f(x) = x² + ax + 1 is increasing on [1, 2]?

Correct Answer

The function f(x) is increasing on [1, 2] for all values of a such that a ≥\geq -2.

Exercise: EXERCISE 6.2 | Q: 14 | (Page 13)
For More Understanding

Explanation

To determine the values of 'a' for which the function is increasing, we analyze the derivative f'(x). According to the principles in the provided context, a function is increasing in an interval if its derivative is non-negative (f'(x) ≥\geq 0) throughout that interval. Since the derivative f'(x) = 2x + a is a linear function, its minimum value on the closed interval [1, 2] occurs at the left endpoint x = 1. By ensuring the derivative is non-negative at this starting point, we guarantee it is non-negative for the entire interval.

Solution Steps

  1. Step 1: Find the derivative of the function. f(x) = x² + ax + 1 f'(x) = 2x + a

  2. Step 2: Apply the condition for an increasing function. For f(x) to be increasing on [1, 2], we must have f'(x) ≥\geq 0 for all x ∈ [1, 2].

  3. Step 3: Analyze the derivative on the interval. Since f'(x) = 2x + a is an increasing linear function, its minimum value on [1, 2] occurs at x = 1.

  4. Step 4: Set up the inequality for the minimum value. f'(1) ≥\geq 0 2(1) + a ≥\geq 0 2 + a ≥\geq 0

  5. Step 5: Solve for a. a ≥\geq -2

PreviousNext