To find the point on the curve where the line tangent to the curve is horizontal, we need to determine where the derivative
is equal to zero. The given curve is described by the equation:
And the derivative is given by:
A horizontal tangent line occurs where the slope of the tangent line is zero. This happens when the numerator of the derivative is zero, since a fraction is zero when its numerator is zero (and its denominator is not zero).So, we set the numerator of the derivative equal to zero and solve for
and
:
Solving for
gives us:
Now, we need to find(s) of
for which this solution is valid. We substitute
back into the original equation to find the corresponding
values:
Substituting
back into the equation
gives us:
Therefore, the point on the curve where the line tangent to the curve is horizontal is \((1, -3)\).