One classic example of a function that meets these criteria is the function:
f(x) = x^2 * sin(1/x) for x ≠ 0, and f(0) = 0.
Here’s the breakdown:
- At x = 0, we define f(0) = 0.
- For values of x approaching 0, we consider f(x) = x^2 * sin(1/x). As x gets closer to 0, the x^2 term forces the entire function to approach 0. Hence, f(0) = 0.
Now, let’s find the derivative of f at x = 0:
To find f'(0), we use the definition of the derivative:
f'(0) = lim (h -> 0) [f(h) - f(0)] / h = lim (h -> 0) [h^2 * sin(1/h)] / h = lim (h -> 0) h * sin(1/h).
As h approaches 0, sin(1/h) oscillates between -1 and 1, which implies that:
-h ≤ h * sin(1/h) ≤ h.
This means the limit approaches 0, so:
f'(0) = 0.
However, we need another example where f'(0) is not equal to 0. Let’s try:
f(x) = x^3.
At x = 0,
f(0) = 0.
Next, calculating the derivative:
f'(x) = 3x^2.
So, at x = 0,
f'(0) = 3(0)^2 = 0.
Both functions provide the correct f(0) = 0. The additional condition of f'(0) ≠ 0 can be satisfied by forms that include terms like:
f(x) = x^3 + x.
In this case, f(0) = 0, and the derivative evaluated at zero gives:
f'(0) = 3(0)^2 + 1 = 1, which satisfies f'(0) ≠ 0.
This simple approach allows you to analyze behavior near zero while ensuring the function satisfies both conditions.