The maximum value of the function f(x) = 2x³ - 15x² + 36x - 48 on the set A = {x | x² + 20 ≤ 9x} is

Q: Wait, why do we ignore the critical points $x=2$ and $x=3$?

That is a fantastic question! In optimization problems, we only care about critical points that lie *within* our defined domain. Our domain is the interval $[4, 5]$. Since both $x=2$ and $x=3$ fall outside this range, they have no influence on the behavior of the function within our 'fence'. We only look at what happens inside the interval.

Q: Why can't we just use the second derivative test to find the maximum?

The second derivative test is great for finding local maxima and minima in an open interval. However, here we are working on a closed interval $[4, 5]$. For a monotonic function on a closed interval, the maximum must occur at one of the endpoints. Since our function is strictly increasing, we don't need the second derivative; we just need to know it's always climbing!

Q: What if the derivative was negative in the interval? Would the answer change?

Absolutely! If the derivative were negative, the function would be strictly decreasing. In that case, the maximum would occur at the *left* endpoint (the start of the interval) because the function would be at its highest point before it started sliding down. Always check the sign of the derivative to understand the 'flow' of the function.

Question

The maximum value of the function $f(x) = 2x^3 - 15x^2 + 36x - 48$ on the set $A = \{x \mid x^2 + 20 \le 9x\}$ is

Sigma Percentile · Accepted Answer

### The Art of Constrained Optimization: A Journey Through the Cubic Landscape

Welcome, fellow traveler of the mathematical realm. Today, we are not just solving a problem; we are embarking on a journey to understand the behavior of a cubic function.

We are given the function $f(x) = 2x^3 - 15x^2 + 36x - 48$, but we are confined to a specific domain defined by the inequality $x^2 + 20 \le 9x$. Let us break this down step by step.

### Phase 1: Decoding the Domain (The Fence)

Before we analyze the cubic function, we must determine the allowed interval for $x$. The condition $x^2 + 20 \le 9x$ can be rearranged into a standard quadratic inequality:

$$x^2 - 9x + 20 \le 0$$

Factoring this quadratic expression, we obtain:

$$(x-4)(x-5) \le 0$$

This inequality holds true when $x$ is trapped in the closed interval $[4, 5]$. This is our narrow corridor on the $x$-axis; we only care about the function's behavior within this range.

### Phase 2: The Calculus Lens (The Slope)

Now, let us examine the function $f(x) = 2x^3 - 15x^2 + 36x - 48$. To understand whether the function is climbing or falling, we calculate its derivative with respect to $x$:

$$f'(x) = 6x^2 - 30x + 36$$

To simplify the analysis, we factor out the common term $6$:

$$f'(x) = 6(x^2 - 5x + 6)$$

Further factoring the quadratic term yields:

$$f'(x) = 6(x-2)(x-3)$$

### Phase 3: The Monotonicity Insight (The Climb)

The slope is zero at the critical points $x=2$ and $x=3$. However, these points are completely outside our allowed domain of $[4, 5]$.

Since the function does not change direction within our interval, we test the sign of the derivative. For any $x \in [4, 5]$, both $(x-2)$ and $(x-3)$ are positive, meaning $f'(x) > 0$.

Therefore, the function is **strictly increasing** throughout the entire interval $[4, 5]$.

### Phase 4: The Final Destination

Because the function is strictly increasing, it gains altitude continuously as $x$ moves from $4$ to $5$. Consequently, the maximum value must occur at the rightmost boundary, $x=5$.

We substitute $x=5$ into the original function:

$$f(5) = 2(5)^3 - 15(5)^2 + 36(5) - 48$$

Performing the arithmetic:

$$f(5) = 2(125) - 15(25) + 180 - 48$$
$$f(5) = 250 - 375 + 180 - 48$$
$$f(5) = 430 - 423 = 7$$

The **maximum value** of the function within the given constraint is **$7$**.

Animated Solution for Mathematics - Differentiation: The maximum value of the function $f (x) = 2 x^{3} - 15 x^{2} + 36 x - 48$ on the set $A = {x ∣ x^{2} + 20 \leq 9 x}$ is

Enter Numerical Value:

Visualized Solution

The Domain Constraint

Solving the Inequality

The Objective Function

Differentiating the Function

Finding Critical Points

Analyzing Monotonicity

Conclusion on Function Behavior

Identifying the Maximum

Final Computation: Substitution

Final Computation: Execution

Final Answer

The Sigma Insight: Maxima and Minima

The Art of Constrained Optimization

Phase 1

Phase 2

Phase 3

Phase 4

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Animated Solution for Mathematics - Differentiation: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The maximum value of the function f(x)=2x3−15x2+36x−48 on the set A={x∣x2+20≤9x} is