Let $f$ be a differentiable function with derivative $f^{\prime }$. Which of the following statements is true?

• A. If $f^{\prime }\left(x\right)=1$ for all $x$, then $f\left(x\right)=x^2$.
• B. If $f^{\prime }\left(x\right)=0$ for all $x$ in the domain of $f$, then $f$ is a constant function.
• C. If $f^{\prime }\left(x\right)>0$ for all $x$, then $f\left(x\right)$ is decreasing everywhere.
• D. If $f^{\prime }\left(x\right)<0$ for all $x$, then $f\left(x\right)$ is constant everywhere.

Answer: (B)

Verified step by step guidance

1. Step 1: Understand the meaning of the derivative f'(x). The derivative represents the rate of change of the function f(x) with respect to x. If f'(x) = 0 for all x, it means the function f(x) does not change as x changes, implying f(x) is constant.
2. Step 2: Analyze the first statement: 'If f'(x) = 1 for all x, then f(x) = x^2.' This is incorrect because the derivative of x^2 is 2x, not 1. If f'(x) = 1, then f(x) must be x + C, where C is a constant.
3. Step 3: Evaluate the second statement: 'If f'(x) = 0 for all x in the domain of f, then f is a constant function.' This is correct because a derivative of 0 indicates no change in the function, meaning f(x) is constant.
4. Step 4: Examine the third statement: 'If f'(x) > 0 for all x, then f(x) is decreasing everywhere.' This is incorrect because a positive derivative indicates that the function is increasing, not decreasing.
5. Step 5: Assess the fourth statement: 'If f'(x) < 0 for all x, then f(x) is constant everywhere.' This is incorrect because a negative derivative indicates that the function is decreasing, not constant.