Question

A real valued function \( f(x)=\left|x^{2}-3x+2\right|+2x-3 \) is defined on \( [-2,1] \). If m and M are absolute minimum and absolute maximum values of \( f \) respectively then \( M-4m= \)

• A. 0
• B. 1
• C. 15
• D. 10

Hint:

Checking the sign of the quadratic inside the modulus is crucial. Since the interval ends at a root (x=1), the expression does not change sign within the interval \([-2, 1)\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:

We first simplify the function by analyzing the sign of the expression inside the modulus over the interval [-2, 1]. Then we find the critical points and evaluate the function at the critical points and endpoints to find the global maximum (M) and minimum (m).

Step 2: Key Formula or Approach:

1. |A| = A if A ≥ 0.
2. Critical points occur where f'(x) = 0 or f'(x) is undefined.
3. Absolute extrema on [a, b] are found by comparing f(a), f(b), and f(c), where c are the critical points.

Step 3: Detailed Explanation:

The expression inside the modulus is:
g(x) = x2 - 3x + 2 = (x - 1)(x - 2)

Roots are at x = 1 and x = 2.

For x ∈ [-2, 1]:
Test x = 0:
g(0) = 2 > 0

So,
x2 - 3x + 2 ≥ 0 on [-2, 1].

Thus,
f(x) = (x2 - 3x + 2) + 2x - 3 = x2 - x - 1

Find the derivative:
f'(x) = 2x - 1

Set f'(x) = 0:
2x - 1 = 0
x = 1/2

This critical point lies in [-2, 1].

Evaluate f(x) at the critical point and endpoints:

1. At x = -2:
f(-2) = (-2)2 - (-2) - 1 = 4 + 2 - 1 = 5

2. At x = 1:
f(1) = 12 - 1 - 1 = -1

3. At x = 1/2:
f(1/2) = (1/2)2 - (1/2) - 1
= 1/4 - 2/4 - 4/4
= -5/4

Comparing these values:
Maximum M = 5
Minimum m = -5/4

Calculate M - 4m:
M - 4m = 5 - 4(-5/4)
= 5 + 5
= 10

Step 4: Final Answer:

The value is 10.