Question

Review of definitions and theorems: State each theorem or definition that follows in precise mathematical language. Then give an illustrative graph or example, as appropriate.

(a) f has a local maximum at x = c .

(b) f has a local minimum at x = c .

(c) f is continuous on [a, b] .

(d) f is differentiable on (a, b) .

(e) The secant line from (a, f (a)) to (b, f (b)) .

(f) The right derivative f ' +(c) at a point x = c .

(g) The left derivative f ' −(c) at a point x = c .

(h) The Extreme Value Theorem .

(i) The Intermediate Value Theorem .

Short answer

The definition and the graph of the given statements follows mathematical language .

Step-by-step solution

Step 1. (a) f has a local maximum at x = c .

A function f has a local maximum at c if there exists an open interval I containing c such that I is contained in the domain of f and f(c) ≥ f(x) for all x ∈ I. A function f has a local minimum at c if there exists an open interval I containing c such that I is contained in the domain of f and f(c) ≤ f(x) for all x ∈ I. A function f has a local extremum at c if f has a local maximum at c or f has a local minimum at c .

Step 2.(b)  f has a local minimum at x = c .

Let f be a function. We say that f has an absolute minimum (or global minimum) at c if f(c) ≤ f(x) for all x in the domain of f. If f has an absolute minimum at c, then f(c) is called the minimum value of f .

Step 3. (c)  f is continuous on [a, b] .

A function is said to be continuous on an interval when the function is defined at every point on that interval and undergoes no interruptions, jumps, or breaks. If some function f(x) satisfies these criteria from x=a to x=b, for example, we say that f(x) is continuous on the interval [a, b].

Step 4. (d) f is differentiable on (a, b) .

A differentiable function is a function that can be approximated locally by a linear function. (c). limit exists for every c ∈ (a, b) then we say that f is differentiable on (a, b). is undefined (0/0) at x = c, but it doesn't have to be defined in order for the limit as x → c exist .

Step 5. (e) The secant line from (a, f (a)) to (b, f (b) .

The slope of a line is defined as rise over run. A secant line of a curve is a line that passes through any two points of the curve. When one of these points is approaching the other, then the slope of the secant line would become the slope of the tangent line at that particular point.

Step 6. (f) The right derivative f '+(c) at a point x = c .

Right hand derivative of a function f(x) at a point x=a are defined as

$f{\text{'}}_{+}\left(c\right)=\underset{h\to 0^{+}}{\lim }\frac{f\left(c+h\right)-f\left(c\right)}{h}$respectively .Let f be a twice differentiable function. We also know that derivative of an even function is odd function and derivative of an odd function is even function .

Step 7. (g) The left derivative f '−(c) at a point x = c .

Left hand derivative of a function f(x) at a point x=a are defined as $f\text{'}\left(c^{-}\right)=\underset{h\to 0^{+}}{\lim }\frac{f\left(c\right)-f\left(c-h\right)}{h}$respectively .Let f be a twice differentiable function. We also know that derivative of an even function is odd function and derivative of an odd function is even function .

Step 8. (h) The Extreme Value Theorem .

If a function f is continuous on [a,b], then it attains its maximum and minimum values on [a,b].

Step (i) The Intermediate Value Theorem .

The intermediate value theorem states that if f is a continuous function whose domain contains the interval [a, b], then it takes on any given value between f(a) and f(b) at some point within the interval.