Describe all of the different ways a function may not be differentiable at the point P (a,f(a)).
There are several ways in which a function may fail to be differentiable at a particular point P (a, f(a)). Here, I will describe the different scenarios:
1. Discontinuity: If a function has a discontinuity at point P, it cannot be differentiable there. Discontinuities occur when there is a jump, hole, or vertical asymptote in the function. For example, the function f(x) = |x| has a discontinuity at x = 0, so it is not differentiable at that point.
2. Sharp Corner or Cusp: If a function has a sharp corner or cusp at point P, it is not differentiable there. This happens when the tangent lines to the function approach different slopes from the left and the right side of the point. For example, the function f(x) = √(|x|) has a sharp corner at x = 0, so it is not differentiable at that point.
3. Vertical Tangent: If a function has a vertical tangent at point P, it is not differentiable there. A vertical tangent occurs when the slope of the tangent line is undefined or approaches infinity. For example, the function f(x) = x^(2/3) has a vertical tangent at x = 0, so it is not differentiable at that point.
4. Horizontal Tangent: If a function has a horizontal tangent at point P, it may or may not be differentiable there. A horizontal tangent occurs when the slope of the tangent line is zero. However, having a horizontal tangent doesn't guarantee differentiability, as there may be other factors at play, such as sharp corners or cusps. For example, the function f(x) = |x| has a horizontal tangent at x = 0, but it is still not differentiable at that point due to the sharp corner.
5. Infinite Oscillation: If a function exhibits infinite oscillation, it may not be differentiable at the point of oscillation. This occurs when the function oscillates infinitely between positive and negative values in a small interval surrounding the point. For example, the function f(x) = sin(1/x) has infinite oscillation around x = 0, so it is not differentiable at that point.
These are some of the different scenarios where a function may not be differentiable at a particular point. To determine the differentiability of a function at a given point, you need to assess if any of these conditions are met by analyzing the behavior of the function in the neighborhood of that point.