
Title: Higher Derivatives

Series: Real Analysis

Chapter: Differentiable Functions

YouTubeTitle: Real Analysis 44  Higher Derivatives

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Quiz: Test your knowledge

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Quiz Content
Q1: Let $f \colon \mathbb{R} \rightarrow \mathbb{R}$ be a function. What is the difference between ‘differentiable’ and ‘continuously differentiable’?
A1: There is no difference.
A2: A differentiable function is always a continuous differentiable function but not vice versa
A3: A continuous differentiable function is a differentiable function where $f^\prime$ is also continuous.
A4: A continuous differentiable function is a differentiable function where $f^\prime$ is also differentiable.
Q2: Let $f \colon \mathbb{R} \rightarrow \mathbb{R}$ be a twotimes differentiable function. Is $f$ continuously differentiable?
A1: Yes!
A2: No!
A3: One cannot say it.
Q3: Let $f \colon \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x) = x^5$. Which claim is not correct?
A1: $f$ is twotimes differentiable.
A2: $f$ is 5times differentiable.
A3: $f$ is 6times differentiable.
A4: $f$ is $\infty$times differentiable.
A5: $f$ has a local maximum at $x_0 = 0$.
Q4: Let $f \colon \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x) = (x1)^2$. Which claim is correct?
A1: $f^{\prime \prime}(0) > 0$ implies there is a local minimum at $x_0 = 0$.
A2: $f^{\prime \prime}(1) > 0$ implies there is a local maximum at $x_0 = 1$.
A3: $f^\prime(1) = 0$ and $f^{\prime \prime}(1) > 0$ imply there is a local minimum at $x_0 = 1$.
A4: $f^\prime(1) = 0$ and $f^{\prime \prime}(1) < 0$ imply there is a local minimum at $x_0 = 1$.