Step 1: Understand the function
The given function is f(x) = x³. This is a cubic function that is continuous and differentiable everywhere.
Step 2: Find the first derivative
Calculate the first derivative to check for stationary points:
f'(x) = 3x²
At x = 0, f'(0) = 0, so x = 0 is a critical point.
Step 3: Find the second derivative
Calculate the second derivative to determine the nature of the critical point:
f''(x) = 6x
At x = 0, f''(0) = 0, so the second derivative test is inconclusive.
Step 4: Analyze the change of concavity
Check the sign of f''(x) around 0:
- For x < 0, f''(x) < 0 (concave down)
- For x > 0, f''(x) > 0 (concave up)
The concavity changes from concave down to concave up as x passes through zero.
Step 5: Conclusion
Since the concavity changes at x = 0, the point is an inflection point, where the curve changes curvature but is neither a local maxima nor minima.
Final Answer: (D) inflexion
Let A be the set of 30 students of class XII in a school. Let f : A -> N, N is a set of natural numbers such that function f(x) = Roll Number of student x.
On the basis of the given information, answer the followingIs \( f \) a bijective function?
Complete and balance the following chemical equations: (a) \[ 2MnO_4^-(aq) + 10I^-(aq) + 16H^+(aq) \rightarrow \] (b) \[ Cr_2O_7^{2-}(aq) + 6Fe^{2+}(aq) + 14H^+(aq) \rightarrow \]