diff --git a/doc/source/whatsnew/v3.0.0.rst b/doc/source/whatsnew/v3.0.0.rst
index 88d36439af1d5..e17d5b0cf8edb 100644
--- a/doc/source/whatsnew/v3.0.0.rst
+++ b/doc/source/whatsnew/v3.0.0.rst
@@ -274,6 +274,7 @@ Performance improvements
 - Performance improvement in :meth:`MultiIndex.equals` for equal length indexes (:issue:`56990`)
 - Performance improvement in :meth:`RangeIndex.__getitem__` with a boolean mask or integers returning a :class:`RangeIndex` instead of a :class:`Index` when possible. (:issue:`57588`)
 - Performance improvement in :meth:`RangeIndex.append` when appending the same index (:issue:`57252`)
+- Performance improvement in :meth:`RangeIndex.argmin` and :meth:`RangeIndex.argmax` (:issue:`57823`)
 - Performance improvement in :meth:`RangeIndex.round` returning a :class:`RangeIndex` instead of a :class:`Index` when possible. (:issue:`57824`)
 - Performance improvement in :meth:`RangeIndex.join` returning a :class:`RangeIndex` instead of a :class:`Index` when possible. (:issue:`57651`, :issue:`57752`)
 - Performance improvement in :meth:`RangeIndex.reindex` returning a :class:`RangeIndex` instead of a :class:`Index` when possible. (:issue:`57647`, :issue:`57752`)
diff --git a/pandas/core/indexes/range.py b/pandas/core/indexes/range.py
index 8199ba8ed3a71..e5e0a4b66f71b 100644
--- a/pandas/core/indexes/range.py
+++ b/pandas/core/indexes/range.py
@@ -515,7 +515,7 @@ def copy(self, name: Hashable | None = None, deep: bool = False) -> Self:
         new_index = self._rename(name=name)
         return new_index
 
-    def _minmax(self, meth: str) -> int | float:
+    def _minmax(self, meth: Literal["min", "max"]) -> int | float:
         no_steps = len(self) - 1
         if no_steps == -1:
             return np.nan
@@ -536,6 +536,39 @@ def max(self, axis=None, skipna: bool = True, *args, **kwargs) -> int | float:
         nv.validate_max(args, kwargs)
         return self._minmax("max")
 
+    def _argminmax(
+        self,
+        meth: Literal["min", "max"],
+        axis=None,
+        skipna: bool = True,
+    ) -> int:
+        nv.validate_minmax_axis(axis)
+        if len(self) == 0:
+            return getattr(super(), f"arg{meth}")(
+                axis=axis,
+                skipna=skipna,
+            )
+        elif meth == "min":
+            if self.step > 0:
+                return 0
+            else:
+                return len(self) - 1
+        elif meth == "max":
+            if self.step > 0:
+                return len(self) - 1
+            else:
+                return 0
+        else:
+            raise ValueError(f"{meth=} must be max or min")
+
+    def argmin(self, axis=None, skipna: bool = True, *args, **kwargs) -> int:
+        nv.validate_argmin(args, kwargs)
+        return self._argminmax("min", axis=axis, skipna=skipna)
+
+    def argmax(self, axis=None, skipna: bool = True, *args, **kwargs) -> int:
+        nv.validate_argmax(args, kwargs)
+        return self._argminmax("max", axis=axis, skipna=skipna)
+
     def argsort(self, *args, **kwargs) -> npt.NDArray[np.intp]:
         """
         Returns the indices that would sort the index and its
diff --git a/pandas/tests/indexes/ranges/test_range.py b/pandas/tests/indexes/ranges/test_range.py
index 635812dcdd9fe..00655f5546df8 100644
--- a/pandas/tests/indexes/ranges/test_range.py
+++ b/pandas/tests/indexes/ranges/test_range.py
@@ -763,6 +763,30 @@ def test_getitem_boolmask_wrong_length():
         ri[[True]]
 
 
+@pytest.mark.parametrize(
+    "rng",
+    [
+        range(0, 5, 1),
+        range(0, 5, 2),
+        range(10, 15, 1),
+        range(10, 5, -1),
+        range(10, 5, -2),
+        range(5, 0, -1),
+    ],
+)
+@pytest.mark.parametrize("meth", ["argmax", "argmin"])
+def test_arg_min_max(rng, meth):
+    ri = RangeIndex(rng)
+    idx = Index(list(rng))
+    assert getattr(ri, meth)() == getattr(idx, meth)()
+
+
+@pytest.mark.parametrize("meth", ["argmin", "argmax"])
+def test_empty_argmin_argmax_raises(meth):
+    with pytest.raises(ValueError, match=f"attempt to get {meth} of an empty sequence"):
+        getattr(RangeIndex(0), meth)()
+
+
 def test_getitem_integers_return_rangeindex():
     result = RangeIndex(0, 10, 2, name="foo")[[0, -1]]
     expected = RangeIndex(start=0, stop=16, step=8, name="foo")