To determine which statement is NOT necessarily true, we analyze the given graph of $f(x)$ and its implications for the derivative $f'(x)$:

Observations:

1. $f'(x)$ is the slope of the tangent to $f(x)$ at any point.
2. Key features:
   • At critical points (peaks and valleys), $f'(x) = 0$.
   • Where $f(x)$ is increasing, $f'(x) > 0$; where $f(x)$ is decreasing, $f'(x) < 0$.

Statements Analysis:

A. $f'(1) \approx 0$:

• At $x = 1$, $f(x)$ has a peak (local maximum).
• Slope of the tangent $= 0$, so $f'(1) = 0$.
• This is true.

B. $f'(-2) > f'(-1)$:

• At $x = -2$, the slope of the tangent is steeply negative.
• At $x = -1$, $f'(x) = 0$ (a critical point at the peak).
• Hence, $f'(-2) < f'(-1)$, making this statement NOT true.

C. $f'(1.5) \approx 0$:

• At $x = 1.5$, $f(x)$ has a valley (local minimum).
• Slope of the tangent $= 0$, so $f'(1.5) = 0$.
• This is true.

D. $f'(0.5) \approx 4$:

• At $x = 0.5$, $f(x)$ appears to increase steeply.
• Slope $f'(x)$ is positive and approximately equal to 4.
• This is true.

Conclusion:

The incorrect statement is B: $f'(-2) > f'(-1)$. In fact, $f'(-2) < f'(-1)$.

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