find an equation of the line that is tangent to the graph of f and parallel to the given line.?

f(x) = 2x²

f '(x) = 4x ←←[gives the slope of the tangent at any point on the curve]

given line is 6x - y + 5 = 0 → y = 6x + 5 has the form y = mx + b where m is the slope

so the slope of the given line is 6

parallel lines have equal slopes

so the slope of the required line is 6

then 4x = 6

x = 6/4 = 3/2

when x = 3/2, f(x) = 2 * (3/2)² = 2 * (9/4) = 9/2

so the required line touches the curve at (3/2, 9/2) with slope 6 and has the form y = mx + b

subsing stuff:

9/2 = 6 * (3/2) + b

9/2 = 9 + b

b = -9/2

so the equation of the required tangent is y = 6x - (9/2)

2y = 12x - 9

12x - 2y = 9

6x-y+5=0

y=6x+5, slope =6

f'(x) =4x

4x=6

x=6/4=3/2

f(3/2) =2(3/2)^2 =2 (9/4) =9/2

at the point (3/2, 9/2), slope 6

y-9/2 =6(x-3/2)

y-9/2 =6x -9

y=6x -9/2