Assertion (A): $f(x) = x^4$ has a local minimum at $x = 0$.

Reason (R): $f''(0) > 0$ for $f(x) = x^4$.

Explanation

1. 01

   $x^4 \ge 0 = f(0)$ for every $x$, so 0 is a minimum: A is true.

2. 02

   $f''(x) = 12x^2$, so $f''(0) = 0$ and R is false. The second derivative test fails here, but the first derivative test ($4x^3$ goes from negative to positive) still finds the minimum.