Can a function be differentiable at a horizontal tangent?
a) Yes
b) No
The Correct Answer and Explanation is :
The answer is:
a) Yes
A function can indeed be differentiable at a horizontal tangent. To understand why, let’s break down what differentiability and a horizontal tangent mean.
- Differentiability: A function is differentiable at a point if it has a defined derivative at that point, meaning the function has a well-defined rate of change at that point. Differentiability implies that the function is smooth and has no sharp corners, jumps, or vertical tangents at the point in question.
- Horizontal Tangent: A horizontal tangent line at a point on the function’s graph means that the slope of the tangent line at that point is zero. In calculus, if ( f(x) ) has a horizontal tangent at a point ( x = c ), then the derivative ( f'(c) = 0 ).
A classic example is the function ( f(x) = x^2 ). The derivative of ( f(x) = x^2 ) is ( f'(x) = 2x ). At ( x = 0 ), the derivative ( f'(0) = 0 ), which indicates a horizontal tangent line at that point. However, the function is differentiable at ( x = 0 ) because the derivative exists and is continuous there.
So, the presence of a horizontal tangent simply indicates that the slope (derivative) is zero at that point, not that the derivative is undefined. Differentiability depends on the function being smooth and having a well-defined derivative, not on the value of the derivative itself. A function can have points where the derivative is zero (horizontal tangents) while still being differentiable at those points.
Therefore, the correct answer is Yes, a function can be differentiable at a point where it has a horizontal tangent.