Question

Multiple Choice If \(f(0)=f^{\prime}(0)=f^{n}(0)=0,\) which of the following must be true? \(\mathrm (A) There is a local maximum of \)f\( at the origin. (B) There is a local minimum of \)f\( at the origin. (C) There is no local extremum of \)f\( at the origin. (D) There is a point of inflection of the graph of \)f\( at the origin. (E) There is a horizontal tangent to the graph of \)f$ at the origin.

Short answer

The correct option is (E). There is a horizontal tangent to the graph of \(f\) at the origin.

Step-by-step solution

Statement Analysis

The given information tells us that \(f(0)=f'(0)=f''(0)=0\). We need to verify, from the options given, which of them must be true based on what we know about the function and its derivatives. The options propose various possibilities of a local maximum, minimum, an inflection point or a horizontal tangent.

Understanding Derivatives

First, let's recall what we know about derivatives. The first derivative of a function at a point gives us the slope of the tangent line to the function at that point. A derivative equal to zero means that the function has a horizontal tangent at that point, indicating a potential maximum, minimum, or inflection point. The second derivative tells us about the concavity of the function. A function has an inflection point (changes concavity) where its second derivative is zero. However, in order for it to be an inflection point, the second derivative must change signs. If the second derivative does not change signs, the function does not actually have an inflection point, even though the second derivative is zero.

Using The Derivative Information

Given that \(f'(0)=0\), we know that the function has a horizontal tangent at \(x=0\), but we cannot determine whether it has a local maximum, local minimum, or point of inflection. Now, with \(f''(0)=0\), it suggests a possible point of inflection at \(x=0\), but we do not have enough information to determine if the concavity changes. Since we don't have enough information to clarify whether it's a local maximum, local minimum or an inflection point we can rule out options (A), (B), and (D).

Final Conclusion

After analyzing each option and applying our understanding of derivatives, we find that option (E) 'There is a horizontal tangent to the graph of \(f\) at the origin' must be true. The other options can't be determined only from the given information.