Question

Find the equation of the line tangent to the graph f(x) = -0.5x3 - 4x2 + 2x – 6 at the point (-6, -54).

Answer 1

To find the slope of the line, we start by calculating the first derivative of the function, as following:

`\begin{gathered} f(x)=-0.5x^3-4x^2+2x-6 \\ \rightarrow f^(\prime)(x)=-1.5x^2-8x+2 \end{gathered}`

The slope of the line tangent to the graph at (-6, -54) is:

`f^(\prime)(-6)=m=-100`

Then, we use the slope-point form to find the equation:

`\begin{gathered} y+54=-100(x+6) \\ \rightarrow y+54=-100x-600 \\ \rightarrow y=-100x-654 \end{gathered}`