find all values of x where the tangent line to the graph of y=gx is parallel to y=15x-30

Question

consider the function

g(x)=(1/3)x^3+2x^2+20x+7

Answer 1

The curve is y = g(x) = (1/3)x³ + 2x² + 20x + 7 and , the tangent line is parallel to y = 15x - 30.

So parallel lines have same slopes.

y = 15x - 30

Compare the above equation with y = mx + c.

Slope  is 15.

Therefore, slope of the tangent line is also 15.

y = (1/3)x³ + 2x² + 20x + 7

Apply derivative on each side.

y ' = x² + 4x + 20

Here y ' is slope of the tangent line.

x² + 4x + 20 = 15

x² + 4x  + 5 = 0

Compare it to .

Roots are

x values are imaginary.

There is no tangent line at real values of x.