Step 1: Definition of a point of inflection.
A point of inflection is a point where the concavity of the function changes. This means \( f''(x) \) changes its sign at that point.
Step 2: Conditions for inflection points.
At a point \( x = c \), the function \( f(x) \) has a point of inflection if: - \( f''(x) \) changes its sign around \( x = c \). - \( f'(x) = 0 \) but does not change its sign.
Step 3: Analyzing the options.
Option (C) correctly states that \( f'(x) = 0 \) and \( f'(x) \) does not change its sign, which is consistent with the definition of a point of inflection.
Step 4: Conclusion.
The correct answer is (C).
Show that \( R \) is an equivalence relation. Also, write the equivalence class \([2]\).
List-I | List-II |
(A) Absolute maximum value | (I) 3 |
(B) Absolute minimum value | (II) 0 |
(C) Point of maxima | (III) -5 |
(D) Point of minima | (IV) 4 |
A battery of emf \( E \) and internal resistance \( r \) is connected to a rheostat. When a current of 2A is drawn from the battery, the potential difference across the rheostat is 5V. The potential difference becomes 4V when a current of 4A is drawn from the battery. Calculate the value of \( E \) and \( r \).