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The problem asks to show that \( f(x(t)) \) is decreasing.

To understand this, let’s break down the steps given in the solution:

1. **Chain Rule Application**:
   \[
   \frac{d}{dt}(f \circ x)(t) = \langle \nabla f(x(t)), \dot{x}(t) \rangle
   \]
   Here, \( f(x(t)) \) is a composition of two functions: \( f \) and \( x(t) \), where \( x(t) \) is a function of \( t \). To differentiate this, we use the **chain rule**. The result is the dot product of the gradient of \( f \) at \( x(t) \) (denoted as \( \nabla f(x(t)) \)) and the derivative of \( x(t) \), which is \( \dot{x}(t) \) (the velocity).

2. **Negative of Norm Squared**:
   \[
   = - \| \nabla f(x(t)) \|^2
   \]
   This step assumes that the vector \( \dot{x}(t) \) is in the **opposite direction** of the gradient \( \nabla f(x(t)) \). This would imply that \( \langle \nabla f(x(t)), \dot{x}(t) \rangle \) equals \( - \|\nabla f(x(t))\|^2 \). Geometrically, this means that the object moves **against** the direction of the gradient, which naturally corresponds to a **decreasing function** because it's going downhill along \( f \).

3. **Conclusion**:
   \[
   \leq 0
   \]
   Since \( \| \nabla f(x(t)) \|^2 \) is always non-negative (as it's a squared norm), the negative sign makes it non-positive, i.e., \( \frac{d}{dt} f(x(t)) \leq 0 \). This shows that \( f(x(t)) \) is a **non-increasing function**.

Thus, the solution demonstrates that \( f(x(t)) \) is decreasing because the time derivative of \( f(x(t)) \) is less than or equal to zero.

---

Would you like further details or clarification on any specific step?

Here are some related questions to deepen your understanding:

1. How is the chain rule applied to a composition of functions in multivariable calculus?
2. What does the gradient \( \nabla f(x(t)) \) represent geometrically?
3. Why does \( \langle \nabla f(x(t)), \dot{x}(t) \rangle = - \|\nabla f(x(t))\|^2 \) assume motion in the opposite direction of the gradient?
4. How do we interpret the condition \( \| \nabla f(x(t)) \|^2 \geq 0 \) in terms of function behavior?
5. What other examples can you think of where a function decreases along the path defined by \( x(t) \)?

**Tip**: In optimization, a function decreases when moving opposite to the gradient direction. This principle is foundational for algorithms like **gradient descent**.

Learn how to demonstrate that f(x(t)) is a decreasing function using multivariable calculus concepts such as the chain rule and gradients. This solution involves calculating the derivative of a composite function and interpreting the result geometrically.

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