The line y = mx + 4 is a tangent to y = 3x^2 + 5x + 7. Find the value of m.

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Answer 1

Equate mx + 4 to 3x² + 5x + 7; then m = (3x² + 5x + 3)/x or 3x + 5 +3/x.

Calculus also gives (for y = 3x² + 5x + 7) dy/dx = 6x + 5, also equal to m.

Then 3x + 5 +3/x = 6x + 5 enables 3/x = 3x or 3x² = 3 or x = ±1.

For x = 1, m = 3x + 5 +3/x = 6x + 5 = 11.

For x = -1, m = 3x + 5 +3/x = 6x + 5 = -1.

A graphing calculator shows both y = 11x + 4 and y = -1x + 4 tangent to

y = 3x² + 5x + 7 at (1,15) and (-1,5), respectively. Both y = 11x + 4 and

y = -x + 4 cross the y-axis at (x,y) = (0,4) in accordance with a mutual

y-intercept of b = 4 (following the lines' common equation form of y = mx + b).