Let a(x) be a differentiable function on the interval [1, ∞), and let ak=a(k) for every positive

step 1 So we're told to let AK be a sequence. We're then asked to prove theorem 7 .6 along with the fol

Let a(x) be a differentiable function on the interval [1,...

Let $a(x)$ be a differentiable function on the interval $[1, \infty)$, and let $a_{k}=a(k)$ for every positive integer $k$. Prove Theorem $7.6$ (c) along with the following variations: (a) Show that when $a^{\prime}(x) \geq 0$ for $x>1$, the sequence $\left\{a_{k}\right\}$ is increasing. (b) Show that when $a^{\prime}(x)>0$ for $x>1$, the sequence $\left\{a_{k}\right\}$ is strictly increasing. (c) Show that when $a^{\prime}(x) \leq 0$ for $x>1$, the sequence $\left\{a_{k}\right\}$ is decreasing. (d) Show that when $a^{\prime}(x)<0$, for $x>1$, the sequence $\left\{a_{k}\right\}$ is strictly decreasing.

Let $a(x)$ be a differentiable function on the interval $[1, \infty)$, and let $a_{k}=a(k)$ for every positive integer $k$. Prove Theorem $7.6$ (c) along with the following variations:
(a) Show that when $a^{\prime}(x) \geq 0$ for $x>1$, the sequence $\left\{a_{k}\right\}$ is increasing.
(b) Show that when $a^{\prime}(x)>0$ for $x>1$, the sequence $\left\{a_{k}\right\}$ is strictly increasing.
(c) Show that when $a^{\prime}(x) \leq 0$ for $x>1$, the sequence $\left\{a_{k}\right\}$ is decreasing.
(d) Show that when $a^{\prime}(x)<0$, for $x>1$, the sequence $\left\{a_{k}\right\}$ is strictly decreasing.

Key Concepts

Sequences Derived from Functions

Sequences defined by sampling a function at integer points, such as a_k = a(k), inherit properties of the original function under appropriate conditions. If the function is monotonic on the domain of interest, then the sequence formed by its evaluations at consecutive integers will similarly be monotonic. This connection allows one to transfer results about the continuous function (derived using calculus) to the discrete sequence, which is often useful in analysis and number theory.

Strict Monotonicity

Strict monotonicity is characterized by the condition where a function not only maintains an increasing or decreasing trend but does so in a manner where no two different points have the same function value. This is ensured by a strictly positive or strictly negative derivative. Strict monotonicity is a stronger form of monotonicity and guarantees uniqueness in order, which among other applications, is crucial in establishing injectivity and the existence of inverses for functions.

Differentiability

Differentiability of a function on an interval means that the function has a well?defined derivative at every point in that interval. This property is essential for applying analytic tools, such as the mean value theorem, which connects the derivative of a function at a point with the overall behavior of the function on intervals. It assures that the function behaves predictably and smoothly, avoiding any abrupt changes in slope or direction.

Derivative Sign and Its Interpretation

The sign of the derivative provides crucial information about the behavior of a function. Specifically, if the derivative a'(x) is nonnegative (a'(x) ? 0), it indicates that the function is nondecreasing, meaning that as x increases, a(x) does not decrease. Similarly, if a'(x) is strictly positive (a'(x) > 0), the function is strictly increasing, implying that an increase in x always results in a strict increase in the value of a(x). The analogous interpretation holds when the derivative is nonpositive or strictly negative, leading to conclusions about the function being nonincreasing or strictly decreasing, respectively.

Monotonicity of Functions

Monotonicity refers to functions that either never decrease or never increase as their independent variable increases. A nondecreasing function maintains the property that a(x1) ? a(x2) when x1 < x2, while a strictly increasing function guarantees a(x1) < a(x2) for x1 < x2. The derivative's sign is the key factor in determining the monotonicity of a differentiable function; hence, when a'(x) maintains a consistent sign over an interval, the function exhibits corresponding monotonic behavior.

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