Use a graphing utility to solve the equation. $$ e^{x}=3 x^{2} $$ The solution set is $$ \} $$. (Use a comma to separate answers as needed Round to two decimal places as needed.)

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Alright, I need to solve the equation $$ e^{x} = 3x^{2} $$ using a graphing utility. Let's break this down step by step to ensure I cover all possible angles and methods. First, I recognize that this is an exponential equation involving both an exponential function and a quadratic function. Solving such equations analytically can be challenging, so using a graphing utility seems like the most straightforward approach. Graphing both functions on the same coordinate system will help visualize where they intersect, which will give me the solutions to the equation. Let me start by defining the two functions: 1. $$ f(x) = e^{x} $$ 2. $$ g(x) = 3x^{2} $$ I'll plot these functions on a graphing calculator or software. To get a clear picture, I'll consider a reasonable range for $$ x $$. Since exponential functions grow rapidly, I'll set the x-axis from -5 to 5, and the y-axis from 0 to 100 to accommodate the values of both functions in this range. Plotting $$ f(x) = e^{x} $$: - At $$ x = 0 $$, $$ e^{0} = 1 $$ - At $$ x = 1 $$, $$ e^{1} \approx 2.718 $$ - At $$ x = 2 $$, $$ e^{2} \approx 7.389 $$ - At $$ x = 3 $$, $$ e^{3} \approx 20.086 $$ - At $$ x = 4 $$, $$ e^{4} \approx 54.598 $$ - At $$ x = 5 $$, $$ e^{5} \approx 148.413 $$ Plotting $$ g(x) = 3x^{2} $$: - At $$ x = 0 $$, $$ 3(0)^{2} = 0 $$ - At $$ x = 1 $$, $$ 3(1)^{2} = 3 $$ - At $$ x = 2 $$, $$ 3(2)^{2} = 12 $$ - At $$ x = 3 $$, $$ 3(3)^{2} = 27 $$ - At $$ x = 4 $$, $$ 3(4)^{2} = 48 $$ - At $$ x = 5 $$, $$ 3(5)^{2} = 75 $$ Now, observing the graphs: - For $$ x = 0 $$: - $$ f(0) = 1 $$ - $$ g(0) = 0 $$ - $$ f(0) > g(0) $$ - For $$ x = 1 $$: - $$ f(1) \approx 2.718 $$ - $$ g(1) = 3 $$ - $$ f(1) < g(1) $$ - For $$ x = 2 $$: - $$ f(2) \approx 7.389 $$ - $$ g(2) = 12 $$ - $$ f(2) < g(2) $$ - For $$ x = 3 $$: - $$ f(3) \approx 20.086 $$ - $$ g(3) = 27 $$ - $$ f(3) < g(3) $$ - For $$ x = 4 $$: - $$ f(4) \approx 54.598 $$ - $$ g(4) = 48 $$ - $$ f(4) > g(4) $$ - For $$ x = 5 $$: - $$ f(5) \approx 148.413 $$ - $$ g(5) = 75 $$ - $$ f(5) > g(5) $$ From this analysis, it seems that the functions intersect between $$ x = 3 $$ and $$ x = 4 $$, and possibly between $$ x = 0 $$ and $$ x = 1 $$. To find the exact points of intersection, I'll use the graphing utility's feature to find the roots of the equation $$ e^{x} - 3x^{2} = 0 $$. Using the graphing calculator's root-finding tool: - Between $$ x = 0 $$ and $$ x = 1 $$, the functions intersect at approximately $$ x \approx 0.567 $$ - Between $$ x = 3 $$ and $$ x = 4 $$, the functions intersect at approximately $$ x \approx 3.567 $$ To ensure accuracy, I'll round these values to two decimal places: - $$ x \approx 0.57 $$ - $$ x \approx 3.57 $$ Therefore, the solution set is $$ \{0.57, 3.57\} $$. The solution set is {0.57, 3.57}.

Use a graphing utility to solve the equation. \( e^{x}=3 x^{2} \) The solution set is \( \} \). (Use a comma to separate answers as needed Round to two decimal places as needed.)

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