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Home » g'(x) = 1 - \frac2x^2. - Coaching Toolbox

g'(x) = 1 - \frac2x^2. - Coaching Toolbox

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 A company's profit \( P \) in thousands of dollars can be modeled by the quadratic equation \( P(x) = -2x^2 + 12x - 16 \), where \( x \) is the number of units produced in hundreds. How many units should the company produce to maximize profit?

📌 The profit function is a quadratic equation in the form \( ax^2 + bx + c \), where \( a = -2 \), \( b = 12 \).

📌 The vertex form of a parabola \( ax^2 + bx + c \) provides the maximum at \( x = -\frac{b}{2a} \).

📌 \[ x = -\frac{12}{2 \times (-2)} = \frac{12}{4} = 3 \]

📌 The company should produce 300 units (since \( x \) is in hundreds) to maximize profit.

📌 The sum of the first \( n \) terms of an arithmetic sequence is given by \( S_n = 3n^2 + 5n \). Find the 10th term of the sequence.

📌 The sum of the first \( n \) terms is \( S_n = 3n^2 + 5n \).

📌 The \( n \)-th term \( a_n \) is given by:

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\text{Diagonal} = 2R = 10 \text{ cm}
For a square with side $ s $, the diagonal is $ s\sqrt{2} $. Therefore...
s\sqrt{2} = 10 \Rightarrow s = \frac{10}{\sqrt{2}} = 5\sqrt{2}
The area of the square is:
s^2 = (5\sqrt{2})^2 = 25 \cdot 2 = 50
Thus, the area is $ \boxed{50} $.
Question: An equilateral triangle has an area of $ 36\sqrt{3} $ cm². I...
The area $ A $ of an equilateral triangle with side length $ s $ is:
A = \frac{\sqrt{3}}{4} s^2
Given $ \frac{\sqrt{3}}{4} s^2 = 36\sqrt{3} $, divide both sides by $...

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