Question

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=3 x^{3}-9 x+1\)

Short answer

The extrema points are at (-1, 13) and (1, -5) and inflection point is at (0,1). Therefore, we sketch a curve that goes through these points. The curve slopes downwards to the left of x = -1, upwards between x = -1 and x = 1, and downwards to the right of x = 1.

Step-by-step solution

Calculate and Identify Extrema

The first step is to find the derivative of \(y = 3x^{3}-9x+1\), which is \(y' = 9x^{2}-9\), and set it equal to zero to identify extrema points.\nSolving the equation \(9x{^2}-9 = 0\) we get \(x = -1, 1\). Calculating the corresponding y-values by substituting these x-values in the original function we get \((-1, 13)\) and \((1, -5)\) respectively.

Calculate and Identify Inflection Points

Next, we find the second derivative of the function, which is \(y'' = 18x\). Set the second derivative equal to zero to find inflection points. Solving \(18x = 0\) gives us \(x = 0\). The corresponding y-value calculated from the original equation is \((0, 1)\). The inflection point is therefore at (0, 1).

Sketch the graph

At last, using all the determined points: extrema points \((-1, 13)\), \((1, -5)\) and inflection point \((0, 1)\) we plot them on the graph. Then we draw smooth, continuous curves that go through each of the points. The curve should be downwards sloping to the left of x = -1, upwards sloping between x = -1 and x = 1, and downwards sloping to the right of x = 1, reflecting the behavior of the cubic function.