1. **Problem Statement:** We are given a function $f$ with a vertical tangent at $x=4$, a horizontal tangent at $x=5$, a jump discontinuity at $x=2$, and a removable discontinuity at $x=7$. We need to identify which statement about differentiability is false. 2. **Recall Differentiability Rules:** - A function is **not differentiable** at points where it has discontinuities (jump or removable). - A function is **not differentiable** at points where the tangent is vertical because the derivative tends to infinity or is undefined. - A function **is differentiable** at points where the tangent is horizontal, as the derivative is zero but defined. 3. **Analyze Each Statement:** - (A) At $x=2$, there is a jump discontinuity, so $f$ is not differentiable there. This is **true**. - (B) At $x=4$, there is a vertical tangent, so $f$ is not differentiable there. This is **true**. - (C) At $x=5$, there is a horizontal tangent. The derivative exists and equals zero, so $f$ is differentiable there. The statement claims $f$ is not differentiable, so this is **false**. - (D) At $x=7$, there is a removable discontinuity, so $f$ is not differentiable there. This is **true**. 4. **Conclusion:** The false statement is (C). **Final answer:** Statement (C) is false because a horizontal tangent at $x=5$ means the function is differentiable there.