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Question #41701

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(t) = t^3 - 6t^2 + 9t$ . Let $S$ be the set of all points $(x, y)$ in the Cartesian plane satisfying the following system of constraints:

$x \ge 0$
$y - x \ge 1$
$2x + y \le 8$
$f(t)$ is strictly increasing on the interval $[x, y]$ .

Define the objective function $Z(x, y) = \frac{f(y) - f(x)}{y - x}$ . Find the maximum value of $Z(x, y)$ subject to $(x, y) \in S$ .

Options

A.
4
B.
9
C.
16
D.
25

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Word Problem Modeling|Increasing/Decreasing Strictness|Mean Value Theorem Applicability

Why This Question Is Tricky

Common Mistakes

This question fuses Linear Programming constraints with Calculus (Function Analysis and Mean Value Theorem).

The core trap lies in the monotonicity constraint: students often mechanically optimize the objective function $Z(x, y)$ over the entire polygonal region defined by the linear inequalities, completely ignoring or misinterpreting the condition that $f(t)$ must be strictly increasing on $[x, y]$ .

If they ignore this, they will maximize $Z(x, y) = x^2 + xy + y^2 - 6(x+y) + 9$ at the vertex $(0, 8)$ , yielding an incorrect answer of 25.

The rigorous path requires identifying that the derivative $f'(t) \ge 0$ restricts the feasible interval to $[0, 1]$ or $[3, \infty)$ .

The LPP constraints then elegantly collapse the feasible region $S$ to a single point $(0, 1)$ , making the optimization trivial once the geometric and calculus constraints are properly synthesized.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Word Problem Modeling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free