What is the value of b in y = 6x^2 + bx + 8 if the curve is tangential to the line y = 2.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The curve y = 6x^2 + bx + 8 is a parabola. The curve is tangential to the line y = 2. This can happen if the parabola opens upwards as well as if it opens downwards.

The slope of a curve y = f(x) at a point where x = a is given by f'(a). Here, the curve is y = 6x^2 + bx + 8. The slope of a horizontal line is equal to 0.

y' = 12x + b = 0

=> x = -b/12

Also, the value of y = 2 where x = -b/12

=> 2 = 6(-b/12)^2 + b(-b/12) + 8

=> 2 = 6*b^2/144 - b^2/12 + 8

=> b^2/24 - b^2/12 + 8 = 2

=> b^2(1/24 - 1/12) = -6

=> b^2(-1/24) = -6

=> b^2 = 6*24

=> b^2 = 144

=> b = 12 and b = - 12

The values of b for which the curve y = 6x^2 + bx + 8 is tangential to y = 2 is b = 12 and b = -12

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