Suppose the graph of a function $f\left(x\right)$ is given below. At which of the following $x$-values is $f\left(x\right)$ NOT differentiable?

• A. At points where the graph is smooth and continuous
• B. At points where the graph has a sharp corner or cusp
• C. At points where the graph has a horizontal tangent
• D. At points where the graph crosses the $x$-axis

Answer: (B)

Verified step by step guidance

1. Step 1: Understand the concept of differentiability. A function is differentiable at a point if it has a well-defined derivative at that point. This means the graph of the function must be smooth and continuous at that point, without any sharp corners, cusps, or vertical tangents.
2. Step 2: Analyze the given options. Differentiability is affected by specific features of the graph: (a) Sharp corners or cusps indicate non-differentiability because the slope changes abruptly. (b) Horizontal tangents do not affect differentiability as the slope is zero but still well-defined. (c) Crossing the x-axis does not inherently affect differentiability unless there is a sharp corner or cusp at the crossing point.
3. Step 3: Identify the key feature that makes a function non-differentiable. The graph is NOT differentiable at points where there are sharp corners or cusps because the derivative does not exist at these points due to the abrupt change in slope.
4. Step 4: Review the other options. Horizontal tangents and crossing the x-axis do not inherently cause non-differentiability unless accompanied by sharp corners or cusps. Smooth and continuous graphs at these points remain differentiable.
5. Step 5: Conclude that the correct answer is: 'At points where the graph has a sharp corner or cusp,' as this is the primary feature that causes non-differentiability in a function.