Question

Prove that y = 6x³+15x+10 has no tangent with slope 12.

Answer 1

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.