A curve has the equation $y = x^3 + 4x^2 + kx + 3$, where $k$ is a constant. Given that the

Question

A curve has the equation $y = x^3 + 4x^2 + kx + 3$, where $k$ is a constant. Given that there exists at least one point on the curve at which the tangent to the curve is perpendicular to the $y$-axis, find the range of values of $k$, and the $x$-coordinate(s), in terms of $k$, of such point(s) on the curve.

Accepted Answer

(i) $k\le\dfrac{16}{3}$ (ii) $x=\dfrac{-4\pm\sqrt{16-3k}}{3}$

Applications of Differentiation - Parameter Conditions for Stationary Points

Applications of Differentiation - Parameter Conditions for Stationary Points

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