if f(x) has a corner or cusp at x = a, then f'(a) is
If a function f(x) has a corner or cusp at x = a, it means that the graph of the function abruptly changes its direction or has a sharp point at x = a
If a function f(x) has a corner or cusp at x = a, it means that the graph of the function abruptly changes its direction or has a sharp point at x = a. In this case, f(x) is not differentiable at x = a because the derivative f'(x) does not exist.
The derivative f'(a) represents the rate of change of the function f(x) at x = a. However, in the case of a corner or cusp, the rate of change is not well defined as there is a sudden change in direction or an undefined slope.
To better understand why f'(a) does not exist at a corner or cusp, let’s consider the definition of the derivative:
f'(a) = lim(h->0) [f(a+h) – f(a)] / h
When there is a corner or cusp, the limit does not exist because the function has different slopes approaching from different directions. Approaching from the left side of the corner or cusp, the slope may be positive, while approaching from the right side, the slope may be negative. Therefore, the limit is not well-defined, and f'(a) is undefined.
In summary, if a function f(x) has a corner or cusp at x = a, then f'(a) does not exist because the function is not differentiable at that point.
More Answers:
Understanding Critical Points in Mathematics | Definition and Analysis TechniquesDifferentiation of g(x) and Finding g'(a) – Math Tutorial for Calculating Derivatives and Substituting Values
Understanding the Relationship Between f'(x) and Local Extrema in Functions