For which value of a is the function f(x) = x^3 - 3ax^2 + 3a^2x + 1 differentiab
Practice Questions
Q1
For which value of a is the function f(x) = x^3 - 3ax^2 + 3a^2x + 1 differentiable at x = 1?
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Questions & Step-by-Step Solutions
For which value of a is the function f(x) = x^3 - 3ax^2 + 3a^2x + 1 differentiable at x = 1?
Step 1: Write down the function f(x) = x^3 - 3ax^2 + 3a^2x + 1.
Step 2: Find the derivative of the function, f'(x).
Step 3: Substitute x = 1 into the derivative to find f'(1).
Step 4: Set f'(1) equal to 0 to find the value of a that makes the function differentiable at x = 1.
Step 5: Solve the equation from Step 4 to find a.
Differentiability – The function must have a defined derivative at the point of interest (x = 1 in this case).
Finding Derivatives – Calculating the derivative of the function and evaluating it at a specific point.
Setting Derivative to Zero – For a function to be differentiable at a point, the derivative at that point must equal zero if looking for critical points.
