91Ó°ÊÓ

A piecewise function is a function defined by different expressions based on the input value's range. For a specific input, only one piece of the function applies.
In the given exercise, the function \(f(x)\) is defined piecewise like this:

• For \(x < 0\): \(f(x) = x^5 \sin\left(\frac{1}{x}\right) + 5x^2\). This part involves a mix of polynomial and trigonometric behavior.
• For \(x = 0\): \(f(x) = 0\). This is a constant value for the point at \(x = 0\).
• For \(x > 0\): \(f(x) = x^5 \cos\left(\frac{1}{x}\right) + \lambda x^2\). Here, \(\lambda\) is an unknown parameter we need to determine.

Each section of the function behaves differently and must be treated separately when evaluating derivatives and limits. Understanding these sections is key when handling continuity across the intervals.

Continuity is a fundamental concept in calculus that ensures a function has no breaks, jumps, or holes at a given point.
For a piecewise function to be fully continuous:

• The function value must be consistent across all pieces at transition points (like at \(x=0\)).
• The derivatives (first and second) should also transition smoothly between different pieces.

In our exercise, we focus on the continuity of \(f'(x)\) and \(f''(x)\) at \(x=0\).
When \(f'(x)\) is examined, it was shown that limits from both sides as \(x\) approaches 0 (\(x \to 0^-\) and \(x \to 0^+\)) were equal to zero. To ensure this, \(2\lambda x\) and \(10x\) needed to match when approaching zero, leading to setting \(\lambda = 5\).
The same value of \(\lambda\) ensures the second derivative \(f''(x)\) is continuous, with both sides of the limit meeting at the value 10 when \(x \to 0\). Every step is essential to guarantee continuity at \(x=0\).

Derivatives measure how a function changes as its input changes. For piecewise functions, we compute derivatives separately in different intervals.

• First Derivative \(f'(x)\): This involves using the product and chain rules to differentiate each segment of \(f(x)\). For \(x < 0\), the derivative was found to include terms like \(5x^4 \sin\left(\frac{1}{x}\right)\), and for \(x > 0\), included \(2\lambda x\).
• Second Derivative \(f''(x)\): We differentiated \(f'(x)\), resulting in terms like \(20x^3 \sin\left(\frac{1}{x}\right)\). Important at \(x=0\) was that the result needed to match for limits from both sides.

Derivatives must be continuous to ensure \(f^{\prime \prime}(0)\) exists, which requires calculating \(\lambda\) as 5. The second derivative explicitly at \(x=0\) checks whether these calculated derivatives smoothly transition across all parts of the function to confirm that there are no interruptions in behavior.