Question

Show that the curve \(y=a x^{3}+b x^{2}+c x+d\) where \(a \neq 0\), has exactly one inflection point.

Short answer

The curve has exactly one inflection point at \(x = -\frac{b}{3a}\).

Step-by-step solution

Locate the necessity for inflection points

An inflection point on a curve is a point where the concavity changes. For a function \(y = f(x)\), the inflection point occurs where the second derivative \(f''(x)\) changes its sign.

Find the first derivative

To find points where concavity changes, the second derivative is needed. Start by finding the first derivative of the curve:\[y = ax^3 + bx^2 + cx + d\]The first derivative \(f'(x)\) is:\[f'(x) = 3ax^2 + 2bx + c\].

Find the second derivative

To identify inflection points, calculate the second derivative:\[f''(x) = 6ax + 2b\]. This derivative tells us about the concavity of the function.

Solve for potential inflection points

Set the second derivative equal to zero to find the critical points for possible inflection:\[6ax + 2b = 0\].Solving for \(x\), we get:\[x = -\frac{b}{3a}\].

Verify the point is an inflection point

To verify that \(x = -\frac{b}{3a}\) is indeed an inflection point, ensure the second derivative changes sign around this point:- When \(x < -\frac{b}{3a}\), \(f''(x) = 6ax + 2b\) will have a different sign than when \(x > -\frac{b}{3a}\).- Since the factor 6ax means this term changes as \(x\) crosses \( -\frac{b}{3a}\), it confirms the change in concavity available at this point.

Key concepts

Concavity

Concavity refers to the way a curve bends. It can either bend upwards or downwards. To determine concavity, we look at the second derivative of a function. A function is said to be *concave up* when its second derivative is positive. This is like the shape of a cup that can hold water. On the other hand, a function is *concave down* when its second derivative is negative, resembling an upside-down cup.
Inflection points occur where the concavity of a function changes. For example, if a graph goes from bending upwards to downwards, it indicates a clear switch in concavity. Recognizing these changes helps us understand the behavior of curves and prepares us for solving questions related to how a function changes over intervals.

Second Derivative

The second derivative of a function helps analyze how the rate of change of the function is itself changing. If you think of driving a car, the first derivative is like your speed, while the second derivative shows whether you are accelerating or braking.

• If the second derivative is positive, it signifies that the function is gaining rate of change and indicates concave up.
• If negative, the function is losing rate of change, indicating concave down.

The importance of the second derivative extends to finding inflection points: places where the second derivative equals zero may indicate a change in the curvature direction. For cubic functions like these, examining the second derivative equips you with the tools to pinpoint such points accurately.

Cubic Functions

Cubic functions are polynomial functions of the form: \(y = ax^3 + bx^2 + cx + d\) where \(a eq 0\). They derive their name from the term involving \(x^3\). These functions can have interesting shapes, including humps or valleys, and often have one inflection point, which is a hallmark of their geometry.
When analyzing a cubic function, finding derivatives helps map out its crest and troughs. The first derivative reveals slopes and extremities, while the second derivative shows positions where the curve switches between bending upwards and downwards.
Understanding these derivatives is crucial in interpreting graph shapes and positions, especially within the real world where such polynomial curves frequently describe everything from economic data to physical paths.

Critical Points

Critical points of a function occur where the first derivative equals zero or is undefined. These are locations where the function could have a hill, valley, or any other kind of stationary point. For inflection points, however, the focus shifts to when the second derivative either equals zero or changes signs.
Finding these points is an essential step when analyzing any function graphically. That's because they signify potential points of interest where the behavior of the function may dramatically change.

• In the context of inflection points, a critical point of interest is established where the second derivative not only becomes zero but also shifts sign.
• This sign change reflects an actual turnaround in concavity, thereby confirming the presence of an inflection point.

Using these concepts effectively guides mathematicians and students in their analysis of functional graphs, underscoring the significance of critical points in the understanding of mathematical behaviors.