Question

give me analysis of a cubic f(x) when f'(x) = 0

Answer 1

The analysis of a cubic function $f(x) = ax^3 + bx^2 + cx + d$ when $f'(x) = 0$ depends on the discriminant of the resulting quadratic equation $3ax^2 + 2bx + c = 0$. Let $\Delta = 4b^2 - 12ac$.

1. If $\Delta > 0$, $f(x)$ has two distinct turning points (one local maximum and one local minimum).
2. If $\Delta = 0$, $f(x)$ has one critical point which is a point of inflection with a horizontal tangent (no local maxima or minima).
3. If $\Delta < 0$, $f(x)$ has no critical points where $f'(x)=0$, meaning it is strictly monotonic (always increasing or always decreasing) and has no local maxima or minima.

Answer 2

For a cubic function $f(x) = ax^3 + bx^2 + cx + d$, where $a \neq 0$, the condition $f'(x) = 0$ is used to find its critical points, which are potential locations for local maxima, local minima, or points of inflection with a horizontal tangent.

1. First Derivative: Calculate the first derivative of the cubic function: $f'(x) = \frac{d}{dx}(ax^3 + bx^2 + cx + d) = 3ax^2 + 2bx + c$

2. Set $f'(x) = 0$: This yields a quadratic equation: $3ax^2 + 2bx + c = 0$ The roots of this quadratic equation are the x-coordinates of the critical points of $f(x)$.

3. Analyze the Discriminant ($\Delta$) of the Quadratic Equation: The discriminant of the quadratic equation $3ax^2 + 2bx + c = 0$ is $\Delta = (2b)^2 - 4(3a)(c) = 4b^2 - 12ac$. The value of $\Delta$ determines the nature of the critical points and, consequently, the shape of the cubic function.

   • Case 1: $\Delta > 0$ (i.e., $4b^2 - 12ac > 0 \implies b^2 - 3ac > 0$)

     • Nature of Critical Points: The quadratic equation $f'(x) = 0$ has two distinct real roots. These two roots correspond to two distinct turning points of the cubic function: one local maximum and one local minimum.
     • Function Behavior: The function changes its direction of monotonicity twice.
       • If $a > 0$, the function increases, then decreases (local maximum), then increases (local minimum).
       • If $a < 0$, the function decreases, then increases (local minimum), then decreases (local maximum).
     • Graph: The graph has the characteristic "S" shape with two distinct "bumps" (turning points).

   • Case 2: $\Delta = 0$ (i.e., $4b^2 - 12ac = 0 \implies b^2 - 3ac = 0$)

     • Nature of Critical Points: The quadratic equation $f'(x) = 0$ has exactly one real root (a repeated root). This root corresponds to a point of inflection where the tangent is horizontal. This point is neither a local maximum nor a local minimum.
     • Function Behavior: The function does not change its direction of monotonicity at this point; its slope momentarily becomes zero.
       • If $a > 0$, the function is strictly increasing (except at the inflection point where the slope is zero).
       • If $a < 0$, the function is strictly decreasing (except at the inflection point where the slope is zero).
     • Graph: The graph has a "stretched S" shape, where the two turning points have merged into a single horizontal inflection point. There are no local maxima or minima.

   • Case 3: $\Delta < 0$ (i.e., $4b^2 - 12ac < 0 \implies b^2 - 3ac < 0$)

     • Nature of Critical Points: The quadratic equation $f'(x) = 0$ has no real roots. This means $f'(x)$ is never zero.
     • Function Behavior: The function is strictly monotonic (it never changes its direction of monotonicity and its slope is never zero).
       • If $a > 0$, the function is strictly increasing for all $x$.
       • If $a < 0$, the function is strictly decreasing for all $x$.
     • Graph: The graph does not have any turning points. It still has an inflection point (where $f''(x) = 0$), but the tangent at this point is not horizontal. There are no local maxima or minima.