Question
If \( f^{\prime}(x)>0 \) then \( f \) is increasing. True False
Ask by Howell Pritchard. in the United States
Feb 26,2025
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Answer
If \( f^{\prime}(x) > 0 \), then \( f \) is increasing.
Solution
The statement "If \( f^{\prime}(x) > 0 \) then \( f \) is increasing" is **True**.
### Explanation:
1. **Definition of Derivative**: The derivative \( f^{\prime}(x) \) represents the rate of change of the function \( f(x) \) at a point \( x \).
2. **Positive Derivative**: If \( f^{\prime}(x) > 0 \), it means that the function \( f(x) \) is increasing at that point. This indicates that as \( x \) increases, \( f(x) \) also increases.
3. **Conclusion**: Therefore, if the derivative is positive over an interval, the function is increasing throughout that interval.
Thus, the statement is true.
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Simplify this solution Beyond the Answer
True! When the derivative \( f'(x) \) is greater than zero, it means that the slope of the tangent line to the function \( f \) at that point is positive. This indicates that as \( x \) increases, \( f(x) \) also increases. So, yes, a function with a positive derivative is indeed increasing over that interval. But don’t confuse this with the possibility of \( f'(x) \) becoming zero or negative at some points—those moments would suggest a flat or decreasing behavior. Keep an eye on those nifty little slopes!
