We are tasked with finding the stationary points for the cubic function $y = 2x^3 - 24x + 13$, which involves finding the critical points and determining whether they are maxima or minima.

Steps to solve:

1. Find the first derivative of the function to locate the stationary points.

   • Given $y = 2x^3 - 24x + 13$, the derivative is: $\frac{dy}{dx} = 6x^2 - 24$

2. Set the derivative equal to zero to find the x-values of the stationary points: $6x^2 - 24 = 0$ Simplifying: $x^2 = 4 \implies x = \pm 2$

3. Find the y-values of the stationary points by substituting $x = 2$ and $x = -2$ into the original equation:

   • For $x = 2$: $y = 2(2)^3 - 24(2) + 13 = 16 - 48 + 13 = -19$ So, one stationary point is $(2, -19)$.
   • For $x = -2$: $y = 2(-2)^3 - 24(-2) + 13 = -16 + 48 + 13 = 45$ So, the other stationary point is $(-2, 45)$.

4. Determine the nature of each stationary point by analyzing the second derivative:

   • The second derivative is: $\frac{d^2y}{dx^2} = 12x$
   • For $x = 2$: $\frac{d^2y}{dx^2} = 12(2) = 24 > 0$ This indicates a local minimum at $(2, -19)$.
   • For $x = -2$: $\frac{d^2y}{dx^2} = 12(-2) = -24 < 0$ This indicates a local maximum at $(-2, 45)$.

Final answer:

• Maximum: $(-2, 45)$
• Minimum: $(2, -19)$

Would you like further details or have any questions?

Related Questions:

1. What does the second derivative tell us about concavity?
2. How do we find inflection points for cubic functions?
3. Why does setting the first derivative equal to zero give stationary points?
4. What is the significance of the sign of the second derivative at a critical point?
5. How would the behavior of the function change if the cubic term had a negative coefficient?

Tip:

Always check the sign of the second derivative to confirm the nature (maximum or minimum) of a stationary point.