нужно найти f'(1)-? f(x)=0 если x=1

Question

f(x)=(x-1)^2*sin1/(x-1)
f(x)=0 если  x=1

Answer 1

To find f'(1), we need to take the derivative of f(x) with respect to x and then evaluate it at x=1.

Let's find the derivative of f(x):
f(x) = (x-1)^2 * sin(1/(x-1))

Using the product rule, we have:
f'(x) = 2(x-1) * sin(1/(x-1)) + (x-1)^2 * cos(1/(x-1)) * (-1/(x-1)^2)

Now, let's evaluate f'(x) at x=1:
f'(1) = 2(1-1) * sin(1/(1-1)) + (1-1)^2 * cos(1/(1-1)) * (-1/(1-1)^2)
= 0 * sin(1/0) + 0 * cos(1/0) * (-1/0)
= 0 + 0 * undefined * undefined

Notice that when x=1, the term sin(1/(x-1)) and cos(1/(x-1)) are undefined since 1/(x-1) results in division by zero. Therefore, we cannot evaluate f'(1) using this function.

Additionally, for f(x) = 0 at x=1, we have:
f(x) = (x-1)^2 * sin(1/(x-1))
0 = (1-1)^2 * sin(1/(1-1))
0 = 0 * sin(1/0)
0 = 0 * undefined

Again, we encounter an undefined term when trying to evaluate f(x) at x=1.

In conclusion, the values of f'(1) and f(1) cannot be determined using the given function.