This is the answer of your question (a) A function f(x) is continuous at a point x if the limit of f(x) as x approaches that point exists and is equal to the value of f(x) at that point. In this case, f(x) is continuous at all values of x because there are no breaks or jumps in the function.
A function f(x) is differentiable at a point x if the derivative of the function exists at that point. The derivative of a function at a point x is given by the limit:
f'(x) = lim h->0 [f(x+h) - f(x)] / h
In this case, f(x) is differentiable at all values of x because the derivative exists for all x.
(b) To find the maximum value of f(x) on the interval [0,10], we need to find the value of x that maximizes f(x). To do this, we can take the derivative of f(x) and set it equal to zero:
f'(x) = 0x (2 - e^(-2u^2)) du + 0 (2 - e^(-2u^2)) dx = 0
This equation simplifies to:
0x du + 0 dx = 0
This equation holds for all values of x, so the maximum value of f(x) on the interval [0,10] occurs at all values of x in that interval.
(c) To find the maximum value of f(x) on the interval [-10,0], we can follow a similar process as in (b). The maximum value of f(x) on the interval [-10,0] also occurs at all values of x in that interval.
This is the answer of your question (a) A function f(x) is continuous at a point x if the limit of f(x) as x approaches that point exists and is equal to the value of f(x) at that point. In this case, f(x) is continuous at all values of x because there are no breaks or jumps in the function.
A function f(x) is differentiable at a point x if the derivative of the function exists at that point. The derivative of a function at a point x is given by the limit:
f'(x) = lim h->0 [f(x+h) - f(x)] / h
In this case, f(x) is differentiable at all values of x because the derivative exists for all x.
(b) To find the maximum value of f(x) on the interval [0,10], we need to find the value of x that maximizes f(x). To do this, we can take the derivative of f(x) and set it equal to zero:
f'(x) = 0x (2 - e^(-2u^2)) du + 0 (2 - e^(-2u^2)) dx = 0
This equation simplifies to:
0x du + 0 dx = 0
This equation holds for all values of x, so the maximum value of f(x) on the interval [0,10] occurs at all values of x in that interval.
(c) To find the maximum value of f(x) on the interval [-10,0], we can follow a similar process as in (b). The maximum value of f(x) on the interval [-10,0] also occurs at all values of x in that interval.
