Let I be an interval containing more than one point and f: I →ℝ be any function. (i) Assume that

step 1 In this problem, we have the left -hand side limit for the derivative at x equals to 0 given as

Let I be an interval containing more than one point and f: I...

Let $I$ be an interval containing more than one point and $f: I \rightarrow \mathbb{R}$ be any function. (i) Assume that $f$ is differentiable. If $f^{\prime}$ is nonnegative on $I$ and $f^{\prime}$ vanishes at only a finite number of points on any bounded subinterval of $I$, then show that $f$ is strictly increasing on $I$. (ii) Assume that $f$ is twice differentiable. If $f^{\prime \prime}$ is nonnegative on $I$ and $f^{\prime \prime}$ vanishes at only a finite number of points on any bounded subinterval of $I$, then show that $f$ is strictly convex on $I$. (iii) Consider $f: \mathbb{R} \rightarrow \mathbb{R}$ given by $f(x)=(x-2 n)^{3}+2 n$, where $n \in \mathbb{Z}$ is such that $x \in[2 n-1,2 n+1) .$ Show that $f$ is differentiable on $\mathbb{R}$ and $f^{\prime \prime}$ exists on $(2 n-1,2 n+1)$, but $f_{+}^{\prime \prime}(2 n+1)=6$, whereas $f_{-}^{\prime \prime}(2 n+1)=-6$ for each $n \in \mathbb{N} .$ Also show that $f$ is strictly increasing on $\mathbb{R}$ although $f^{\prime}(2 n)=0$ for each $n \in \mathbb{N}$. (Compare (i) above and Exercise 12 in the list of Revision Exercises at the end of Chapter 7.) (iv) Consider $g: \mathbb{R} \rightarrow \mathbb{R}$ given by $g(x)=(x-2 n)^{4}+8 n x$, where $n \in \mathbb{Z}$ is such that $x \in[2 n-1,2 n+1) .$ Show that $g$ is twice differentiable on $\mathbb{R}$ and $g^{\prime \prime \prime}$ exists on $(2 n-1,2 n+1)$, but $g_{+}^{\prime \prime \prime}(2 n+1)=24$, whereas $g_{-}^{\prime \prime \prime}(2 n+1)=-24$ for each $n \in \mathbb{N} .$ Also show that $g$ is strictly convex on $\mathbb{R}$ although $g^{\prime \prime}(2 n)=0$ for each $n \in \mathbb{N}$. (Compare (ii) above and Exercise 13 in the list of Revision Exercises at the end of Chapter $7 .$ )

Let $I$ be an interval containing more than one point and $f: I \rightarrow \mathbb{R}$ be any function.
(i) Assume that $f$ is differentiable. If $f^{\prime}$ is nonnegative on $I$ and $f^{\prime}$ vanishes at only a finite number of points on any bounded subinterval of $I$, then show that $f$ is strictly increasing on $I$.
(ii) Assume that $f$ is twice differentiable. If $f^{\prime \prime}$ is nonnegative on $I$ and $f^{\prime \prime}$ vanishes at only a finite number of points on any bounded subinterval of $I$, then show that $f$ is strictly convex on $I$.
(iii) Consider $f: \mathbb{R} \rightarrow \mathbb{R}$ given by $f(x)=(x-2 n)^{3}+2 n$, where $n \in \mathbb{Z}$ is such that $x \in[2 n-1,2 n+1) .$ Show that $f$ is differentiable on $\mathbb{R}$ and $f^{\prime \prime}$ exists on $(2 n-1,2 n+1)$, but $f_{+}^{\prime \prime}(2 n+1)=6$, whereas $f_{-}^{\prime \prime}(2 n+1)=-6$
for each $n \in \mathbb{N} .$ Also show that $f$ is strictly increasing on $\mathbb{R}$ although $f^{\prime}(2 n)=0$ for each $n \in \mathbb{N}$. (Compare (i) above and Exercise 12 in the list of Revision Exercises at the end of Chapter 7.)
(iv) Consider $g: \mathbb{R} \rightarrow \mathbb{R}$ given by $g(x)=(x-2 n)^{4}+8 n x$, where $n \in \mathbb{Z}$ is such that $x \in[2 n-1,2 n+1) .$ Show that $g$ is twice differentiable on $\mathbb{R}$ and $g^{\prime \prime \prime}$ exists on $(2 n-1,2 n+1)$, but $g_{+}^{\prime \prime \prime}(2 n+1)=24$, whereas
$g_{-}^{\prime \prime \prime}(2 n+1)=-24$ for each $n \in \mathbb{N} .$ Also show that $g$ is strictly convex on $\mathbb{R}$ although $g^{\prime \prime}(2 n)=0$ for each $n \in \mathbb{N}$. (Compare (ii) above and Exercise 13 in the list of Revision Exercises at the end of Chapter $7 .$ )

Key Concepts

Differentiability

Differentiability is a fundamental concept in calculus that describes whether a function has a derivative at each point in its domain. It indicates that the function is locally approximable by a linear function, which is essential in understanding the behavior and changes of the function, such as rates of increase and curvature.

Monotonicity and Strictly Increasing Functions

Monotonicity refers to the property of a function preserving order, and a function is strictly increasing if each increase in the input results in a strict increase in the output. A common criterion is that if a function’s derivative is positive, the function is strictly increasing. The analysis may also involve exceptions where the derivative is zero only at finitely many points without affecting the overall strict increase.

Convexity and Strict Convexity

Convexity describes the shape of a function's graph, with a convex function having the property that the line segment connecting any two points on the graph lies above or on the graph. A function is strictly convex if this condition holds in a strict sense, often ensured by a positive second derivative, which indicates that the function's rate of increase is itself increasing.

Behavior of Derivatives with Isolated Zero Points

When a function's derivative (first or second) is nonnegative and only equals zero at finitely many points in any bounded interval, it reinforces strict monotonicity or strict convexity. This assumption is crucial because, even if the derivative is zero at some isolated points, it does not prevent the overall increasing or convex nature of the function.

Piecewise Function Analysis and One-Sided Derivatives

Analyzing piecewise-defined functions involves careful consideration of their behavior at the junctions between pieces. Even though a function might be differentiable or twice differentiable overall, one-sided derivatives such as right-hand and left-hand derivatives can differ at the points where the function definition changes. This analysis is essential for understanding the nuances of differentiability and curvature in functions that are defined in pieces.

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