diff --git a/pyproject.toml b/pyproject.toml
index ae89133..24276c9 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -46,7 +46,8 @@ testing = [
     "coverage>=7.4.3",
     "hypothesis>=6.99.11",
     "mypy>=1.9.0",
-    "pytype>=2023.12.18"
+    "pytype>=2023.12.18",
+    "logot"
 ]
 dev = [
     "bokeh>=3.3.2",
diff --git a/src/depiction/spectrum/peak_picking/basic_interpolated_peak_picker.py b/src/depiction/spectrum/peak_picking/basic_interpolated_peak_picker.py
index c961ef5..f904325 100644
--- a/src/depiction/spectrum/peak_picking/basic_interpolated_peak_picker.py
+++ b/src/depiction/spectrum/peak_picking/basic_interpolated_peak_picker.py
@@ -67,7 +67,7 @@ def _interpolate_max_mz_and_intensity(
         # Fit a cubic spline
         spline = scipy.interpolate.CubicSpline(interp_mz, interp_int)
         roots = spline.derivative().roots()
-        if len(roots) == 0:
+        if np.isnan(roots).any():
             loguru.logger.warning(
                 f"Error: {len(roots)} roots found for local maximum at index {local_max_index}; "
                 f"{interp_mz=}, {interp_int=}, {roots=}"
@@ -89,3 +89,7 @@ def _find_local_maxima_indices(self, mz_arr: NDArray[float], int_arr: NDArray[fl
             ),
         )
         return local_maxima_indices
+
+
+# TODO the interpolation could be much faster, if it were implemented in numba for our specific case of 3 points,
+#      since in general the scipy library will do everything much moe general than is actually required.
diff --git a/tests/unit/spectrum/peak_picking/test_basic_interpolated_peak_picker.py b/tests/unit/spectrum/peak_picking/test_basic_interpolated_peak_picker.py
index 96f880f..509cad2 100644
--- a/tests/unit/spectrum/peak_picking/test_basic_interpolated_peak_picker.py
+++ b/tests/unit/spectrum/peak_picking/test_basic_interpolated_peak_picker.py
@@ -3,6 +3,8 @@
 from unittest.mock import patch, MagicMock
 
 import numpy as np
+from logot import Logot, logged
+from logot.loguru import LoguruCapturer
 
 from depiction.spectrum.peak_picking import BasicPeakPicker, BasicInterpolatedPeakPicker
 
@@ -23,11 +25,46 @@ def basic_interpolated_peak_picker(self) -> BasicInterpolatedPeakPicker:
             peak_filtering=self.mock_peak_filtering,
         )
 
-    def test_interpolate_max_mz_and_intensity_when_success(self) -> None:
-        pass
+    def test_interpolate_max_mz_and_intensity_when_success_and_exact(self) -> None:
+        """In this very simple case, interpolation should return the same value as the intensity on both sides is
+        symmetric."""
+        local_max_index = 2
+        mz_arr = np.array([1., 2, 3, 4, 5])
+        int_arr = np.array([0., 0, 10, 0, 0])
+
+        interpolated_mz, interpolated_int = self.basic_interpolated_peak_picker._interpolate_max_mz_and_intensity(
+            local_max_index, mz_arr, int_arr
+        )
+
+        self.assertAlmostEqual(3, interpolated_mz)
+        self.assertAlmostEqual(10, interpolated_int)
+
+    def test_interpolate_max_mz_and_intensity_when_success_and_not_exact(self) -> None:
+        local_max_index = 2
+        mz_arr = np.array([1., 2, 3, 4, 5])
+        int_arr = np.array([0., 0, 10, 5, 0])
+
+        interpolated_mz, interpolated_int = self.basic_interpolated_peak_picker._interpolate_max_mz_and_intensity(
+            local_max_index, mz_arr, int_arr
+        )
+
+        self.assertAlmostEqual(3.16667, interpolated_mz, places=5)
+        self.assertAlmostEqual(10.20833, interpolated_int, places=5)
 
     def test_interpolate_max_mz_and_intensity_when_failure(self) -> None:
-        pass
+        local_max_index = 2
+        mz_arr = np.array([1., 2, 3, 4, 5])
+        int_arr = np.array([0., 10, 10, 10, 0])
+
+        with Logot().capturing(capturer=LoguruCapturer) as logot:
+            interpolated_mz, interpolated_int = self.basic_interpolated_peak_picker._interpolate_max_mz_and_intensity(
+                local_max_index, mz_arr, int_arr
+            )
+        self.assertIsNone(interpolated_mz)
+        self.assertIsNone(interpolated_int)
+        logot.assert_logged(logged.warning("Error: %d roots found for local maximum at index 2; %s"))
+
+    # TODO test interpolate when there are 2 roots because of numerical issues, 3 is apparently also possible
 
     @patch.object(BasicPeakPicker, "get_min_distance_indices")
     def test_find_local_maxima_indices(self, mock_get_min_distance_indices) -> None: