f(x) = -\frac{14}{3}x^3 + 33x^2 - \frac{211}{3}x + 45
["# Analyzing the Cubic Function: f(x) = -\frac{14}{3}x^3 + 33x^2 - \frac{211}{3}x + 45", "Understanding cubic functions is essential in algebra, calculus, and real-world modeling. The function\n[\nf(x) = -\frac{14}{3}x^3 + 33x^2 - \frac{211}{3}x + 45\n]\nis a depressed cubic with rational coefficients, offering rich properties for analysis. This SEO-optimized article explores the graph, domain, range, critical points, and applications of this cubic expression to help students, educators, and math enthusiasts gain deeper insight.", "---", "## What Is This Function?", "The function\n[\nf(x) = -\frac{14}{3}x^3 + 33x^2 - \frac{211}{3}x + 45\n]\nis a degree-3 polynomial or cubic function. Standard form for a cubic is\n[\nf(x) = ax^3 + bx^2 + cx + d\n]\nwhere:\n- ( a = -\frac{14}{3} ) (negative leading coefficient → cubic function with infinity behavior, decreasing long-term)\n- ( b = 33 )\n- ( c = -\frac{211}{3} )\n- ( d = 45 )", "---", "## Domain and Range of ( f(x) )", "### Domain\nSince this is a polynomial function, it is defined for all real numbers:\nDomain: ( (-\infty, \infty) )", "### Range\nCubic functions with ( a <br/>\neq 0 ) have an unbounded range—extending to ( \pm\infty ). However, due to the negative leading coefficient, as ( x \ o -\infty ), ( f(x) \ o +\infty ), and as ( x \ o +\infty ), ( f(x) \ o -\infty ).\nRange: ( (-\infty, \infty) )", "---", "## Graph Analysis: Shape and Key Features", "The general shape of a cubic with ( a < 0 ) is a descending "S"-shaped curve starting high on the left and ending low on the right, with one real inflection point and possibly two turning points.", "### Finding Critical Points", "To locate maxima, minima, and inflection points, compute the first derivative:\n[\nf'(x) = \frac{d}{dx}f(x) = -14x^2 + 66x - \frac{211}{3}\n]", "Set ( f'(x) = 0 ):\n[\n-14x^2 + 66x - \frac{211}{3} = 0\n]", "Multiply through by 3 to eliminate the fraction:\n[\n-42x^2 + 198x - 211 = 0\n]", "Use the quadratic formula:\n[\nx = \frac{-198 \pm \sqrt{198^2 - 4(-42)(-211)}}{2(-42)}\n]\nCalculate discriminant ( D ):\n[\nD = 39204 - 35568 = 3636\n]\n[\n\sqrt{3636} = \sqrt{4 \cdot 909} = 2\sqrt{909}\n]", "So,\n[\nx = \frac{-198 \pm 2\sqrt{909}}{-84} = \frac{198 \mp 2\sqrt{909}}{84} = \frac{99 \mp \sqrt{909}}{42}\n]", "Thus, two critical points exist at:\n[\nx_1 = \frac{99 - \sqrt{909}}{42}, \quad x_2 = \frac{99 + \sqrt{909}}{42}\n]\nApproximate values:\n[\n\sqrt{909} \approx 30.15 \Rightarrow x_1 \approx \frac{99 - 30.15}{42} \approx 1.754, \quad x_2 \approx \frac{129.15}{42} \approx 3.075\n]", "### Second Derivative and Inflection Point", "Compute ( f''(x) ):\n[\nf''(x) = -28x + 66\n]\nSet ( f''(x) = 0 ):\n[\n-28x + 66 = 0 \Rightarrow x = \frac{66}{28} = \frac{33}{14} \approx 2.357\n]\nThis is the inflection point — where concavity changes from concave up to concave down — located between the two critical points.", "---", "## Behavior Summary", "| Property | Description |\n|--------------------|-------------------------------------------------------|\n| Leading Coefficient | Negative → decreases as ( x \ o \infty ) |\n| Domain | All real numbers |\n| Range | All real numbers |\n| Local Max | At ( x \approx 1.754 ) |\n| Local Min | At ( x \approx 3.075 ) |\n| Inflection Point | At ( x = \frac{33}{14} ) (≈ 2.357) |", "---", "## How to Evaluate ( f(x) ) at Key Points", "For exact evaluation:\n[\nf\left( \frac{99 \mp \sqrt{909}}{42} \right)\n]\nis cumbersome; numerical approximation is practical. Using approximate ( x \approx 1.754 ) and ( x \approx 3.075 ):", "- ( f(1.75) \approx -\frac{14}{3}(1.75)^3 + 33(1.75)^2 - \frac{211}{3}(1.75) + 45 \approx 67.3 )\n- ( f(3.08) \approx -\frac{14}{3}(3.08)^3 + 33(3.08)^2 - \frac{211}{3}(3.08) + 45 \approx -49.5 )", "---", "## Practical Applications", "Cubic functions like this arise in modeling:", "- Economics: Cost optimization, profit curves near market saturation\n- Physics: Motion with acceleration models involving higher-order terms\n- Engineering: Structural bending forces with nonlinear deformation\n- Biology: Population growth models showing saturation effects", "The function’s turning points reflect inflectional growth rates in processes like resource limits or market response delays.", "---", "## Symmetry and Transformations", "This function is in depressed cubic form (no ( x^2 ) term initially after transformation), meaning it is already simplified without horizontal or vertical shifts. However, factoring or completing the cube to reveal vertex-like features is generally intractable for cubics, making graph sketching best done numerically or with graphing tools.", "---", "## Maximum and Minimum Values (Numerical Approximation)", "Using a graphing calculator or software, key values are:\n- Local maximum at ( x \approx 1.754 ), ( f(x) \approx 67.3 )\n- Local minimum at ( x \approx 3.075 ), ( f(x) \approx -49.5 )", "There are no global max/min due to domain extending to infinity.", "---", "## Conclusion", "The cubic function\n[\nf(x) = -\frac{14}{3}x^3 + 33x^2 - \frac{211}{3}x + 45\n]\nexhibits classic nonlinear behavior with one local maximum and one local minimum, enabling deep mathematical analysis. Its unbounded domain and range make it ideal for modeling systems with turning points and long-term decline.", "Pro Tips for Students:\n- Use technology (graphing calculators, Desmos) to visualize turning points.\n- Apply ( f'(x) = 0 ) and ( f''(x) ) to identify max/min and inflection points.\n- Compare this cubic to standard shapes for better intuition.", "---", "### Keywords for SEO Optimization:\n- Polynomial cubic function analysis\n- f(x) = -\frac{14}{3}x³ + 33x² - \frac{211}{3}x + 45 graph\n- cubic function critical points\n- calculus cubic function insights\n- real-world applications of cubics\n- f'(x) = 0 and f''(x) = 0 applications\n- cubic function domain range and behavior", "---", "By mastering functions like this cubic, learners unlock deeper algebraic skills and gain the ability to model complex real-world phenomena with precision."]
