Understanding the Concept of f'(x) > 0: Function Increase and Positive Derivative

f'(x)>0

When we say that f'(x) is greater than 0, we are referring to the derivative of the function f(x)

When we say that f'(x) is greater than 0, we are referring to the derivative of the function f(x).

If f'(x) is greater than 0, it means that the slope of the function f(x) is positive for all values of x. In other words, the function is increasing for all values of x.

To understand this concept further, let’s consider an example:

Suppose we have a function f(x) = x^2. To find the derivative, we differentiate the function with respect to x. In this case, the derivative of f(x) would be f'(x) = 2x.

Now, we need to determine when f'(x) is greater than 0. To do this, we set up the inequality 2x > 0 and solve for x.

Dividing both sides of the inequality by 2, we have x > 0.

This means that for the function f(x) = x^2, the derivative f'(x) is greater than 0 when x is greater than 0.

Graphically, this can be represented by a graph of the function f(x) = x^2, where the curve increases as we move from left to right, indicating that the derivative is positive.

In summary, when f'(x) is greater than 0, it means that the function f(x) is increasing for all values of x.

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