Prove that y = 6x³+15x+10 has no tangent with slope 12.
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we have to prove that, y = 6x³ + 15x + 10 has no tangent with slope 12.
concept : if a given curve is in the form of ,y = f(x).
slope of tangent of the curve = dy/dx .
so, first differentiate y with respect to x,
i.e., dy/dx = d(6x³ + 15x + 10)/dx
⇒dy/dx = d(6x³)/dx + d(15x)/dx + d(10)/dx
⇒dy/dx = 18x² + 15 + 0
⇒dy/dx = 18x² + 15
hence, slope of tangent of the curve is 18x² + 15.
if we assume, 12 is slope of tangent of the curve.
then, 18x² + 15 = 12
or, 18x² = 12 - 15 = -3
or, 18x² = -3
here you see, LHS is always positive term while RHS is negative term.
so, LHS ≠ RHS.
our assumption is wrong that slope of tangent of the curve is 12.
hence, it is clear that y = 6x³ + 15x + 10 has no tangent with slope 12.
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