Question

1. the equation of a curve is ( y=-x^{3}+6 x^{2}-11 x + 6 ). a section of the curve is shown on the diagram below, with the three ( x ) - intercepts labelled.

a) find ( \frac{d y}{d x} ).
b) write down the coordinates of the local maximum.

Explanation:

Step1: Differentiate the function

Use the power rule \(\frac{d}{dx}(x^n)=nx^{n - 1}\).
For \(y=-x^{3}+6x^{2}-11x + 6\), we have:
\(\frac{dy}{dx}=-3x^{2}+12x-11\)

Step2: Analyze the graph for local maximum

From the graph, we observe the \(x\) - coordinate of the local maximum.
Since the \(x\) - intercept at \(x = 2\) is given and by analyzing the shape of the cubic function \(y=-x^{3}+6x^{2}-11x + 6\) (the coefficient of \(x^{3}\) is negative, so the function goes from \(-\infty\) to \(+\infty\) as \(x\) goes from \(+\infty\) to \(-\infty\) in a non - standard cubic way due to the negative leading coefficient).
By further inspection of the graph (assuming basic graph - reading skills for a cubic function), the \(x\) - value of the local maximum is \(x = 3\). Substitute \(x = 3\) into \(y=-x^{3}+6x^{2}-11x + 6\):
\(y=-(3)^{3}+6(3)^{2}-11(3)+6=-27 + 54-33 + 6=0\)

Answer:

a) \(\frac{dy}{dx}=-3x^{2}+12x - 11\)
b) \((3,0)\)