Please help with the questions in the image

We do this by differentiating the polynomials. There are a few methods to do this but I am going to use the power rule, which we multiply the constant by the exponent on the variable and subtract one from the exponent.

[tex]\frac{d}{dx}(-2x^3 + 5x^2 + 9)[/tex]

[tex]-6x^2 + 10x [/tex]

[tex]m = -6x^2 + 10x [/tex] when x = a

Now that we have this information, we can answer question b

Question b

The tangent line for Point (1, 12)

First find the slope by using our derivative.

[tex]m = -6x^2 + 10x [/tex]

[tex]m = -6(1)^2 + 10(1) [/tex]

[tex]m = -6 + 10x [/tex]

[tex]m = 4 [/tex]

Now that we have our slope, use point slope form to find our tangent line

[tex]y - 12 = 4(x - 1)[/tex]

[tex]y(x) = 4x + 8[/tex]

Now lets do the same for the Point (2, 13)

Find the slope at the point.

[tex]m = -6x^2 + 10x [/tex]

[tex]m = -6(2)^2 + 10(2) [/tex]

[tex]m = -24 + 10 [/tex]

[tex]m = -4 [/tex]

Now find the tangent line using point slope form of a line.

[tex]y(x) - 13 = -4(x - 2)[/tex]

[tex]y(x) = -4x + 21[/tex]

Now graph the lines, which I have done and you can see by viewing the image I have attached.