Verified Answer :
- Part 1: Inflection points for \(y=x^4+ax^3+3ax^2+1\) require the quadratic \(2x^2+ax+a\) to have two distinct roots; solving the discriminant \(a^2-8a>0\) gives \(\boxed{a < 0 \text{ or } a > 8}\).
- Part 2: Inflection points for \(y=(x^2+2x+a)e^x\) depend on the discriminant \(12-4a\) of \(x^2+6x+a+6\). There are \(\boxed{2}\) inflection points if \(a < 3\) and \(\boxed{0}\) if \(a \geq 3\).
Explanation
- Inflection Points for the Quartic Curve
To find the range of constant \(a\) for which the curve \(y = x^4 + ax^3 + 3ax^2 + 1\) has inflection points, we first calculate the second derivative.
The first derivative is given as \(y' = 4x^3 + 3ax^2 + 6ax\).
Differentiating again, we get:
\[y'' = 12x^2 + 6ax + 6a = 6(2x^2 + ax + a)\]
An inflection point occurs where the second derivative equals zero and changes sign. For the quadratic \(2x^2 + ax + a = 0\) to have two distinct real roots (where the sign change will occur), its discriminant \(D\) must be strictly greater than zero.
Calculating the discriminant:
\[D = a^2 - 4(2)(a) = a^2 - 8a\]
Setting \(D > 0\):
\[a^2 - 8a > 0 \implies a(a - 8) > 0\]
The roots of the equation \(a(a-8)=0\) are \(a=0\) and \(a=8\). The inequality \(a(a-8) > 0\) holds when \(a < 0\) or \(a > 8\).
Thus, the curve has inflection points for \(a \in (-\infty, 0) \cup (8, \infty)\).
- Inflection Points for the Exponential-Polynomial Curve
To find the number of inflection points for the curve \(y = (x^2 + 2x + a)e^x\), we need to determine where the second derivative changes sign.
First, compute the first derivative using the product rule:
\[y' = (2x + 2)e^x + (x^2 + 2x + a)e^x = (x^2 + 4x + a + 2)e^x\]
Next, compute the second derivative:
\[y'' = (2x + 4)e^x + (x^2 + 4x + a + 2)e^x = (x^2 + 6x + a + 6)e^x\]
Since \(e^x\) is always positive, the sign of \(y''\) depends entirely on the quadratic \(Q(x) = x^2 + 6x + a + 6\).
The number of inflection points depends on the discriminant \(D\) of \(Q(x)\):
\[D = 6^2 - 4(1)(a + 6) = 36 - 4a - 24 = 12 - 4a\]
- If \(D > 0\) (\(a < 3\)), there are two distinct real roots. The quadratic changes sign at both roots, resulting in 2 inflection points.
- If \(D = 0\) (\(a = 3\)), there is one repeated root. The quadratic does not change sign at a repeated root, so there are 0 inflection points.
- If \(D < 0\) (\(a > 3\)), there are no real roots, so there are 0 inflection points.
- Final Answer Summary
The range of values for \(a\) in the first part is \(a < 0\) or \(a > 8\).
For the second part, the number of inflection points is 2 if \(a < 3\) and 0 if \(a \geq 3\).
Examples
In structural engineering, identifying inflection points on curved beams or supports is critical for analyzing bending moments. If a bridge deck is designed with a specific curvature profile like \(y = (x^2 + 2x + a)e^x\), the inflection points indicate where the bending moment changes from tension to compression. By calculating these points mathematically, engineers ensure the structural integrity of the bridge under varying loads, preventing material fatigue or failure.