Q. 1 TB - Calculus

Question

To state the given definitions and theorems.

Step-by-Step Solution

Answer

Extreme Value Theorem: A real valued function$f$must reach a maximum and a minimum at least once if it is continuous on a closed interval.

The intermediate value theorem states that if a real valued function $f$ is continuous on a closed interval$[a, b]$, then at some point in the interval  $[a, b]$, $f$ can take any value between $f\left(a\right)$and $f\left(b\right)$.

Step 1: Given Information

• At $x=c, f$has a local maximum.
• At$x=c, f$ has a local minimum.
• On $[a, b] f$ is continuous.
• $f$ is distinguishable on $(a, b)$
• From $(a, f(a\left)\right)$to $(b, f(b\left)\right)$, the secant line
• The right derivative $f^{\text{'}}+\left(c\right)$ at a point $x=c$
• The left derivative $f^{\text{'}}-\left(c\right)$ at a point $x=c$
• Extreme Value Theorem
• The Intermediate Value Theorem

Step 2: Calculation

• If some, then$f$ has a local maximum at $x=c$ $\delta >0$ like that $f\left(c\right)\ge f\left(x\right)\forall x\in (c-\delta ,c+\delta )$
• If some, then f has a local minimum at x=c $\delta >0$like that $f\left(c\right)\le f\left(x\right)\forall x\in (c-\delta ,c+\delta )$
• If $f$is continuous on all points in the interval $(a, b)$, as well as at right and left continuous at $a$ and$b$, then $f$ is said to be continuous on $[a, b]$.
• If $f$is continuous throughout the entire interval $(a, b)$ , then it is said to be differentiable on  $(a, b)$,

$$\begin{gathered} f^{\text{'}}\left(x\right)=\underset{h\to 0^{-}}{\lim }\frac{f(x+h)-f\left(x\right)}{h} \\ =\underset{h\to 0^{-}}{\lim }\frac{f(x-h)-f\left(x\right)}{-h} \end{gathered}$$

Step 3: Calculation

• The equation yields the secant line $\frac{yf\left(a\right)}{x-a}=\frac{f\left(b\right)-f\left(a\right)}{b-a}$
• At a position where $x=c$, the right derivative is given by $f^{\text{'}}+\left(c\right)=\underset{x\to c^{+}}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}$
• At a position where $x=c$, the left derivative is given by $f^{\text{'}}-\left(c\right)=\underset{x\to c^{-}}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}$
• Extreme Value Theorem: A real valued functionmust reach a maximum and a minimum at least once if it is continuous on a closed interval.
• The intermediate value theorem states that if a real valued function  is continuous on a closed interval, then at some point in the interval  ,  can take any value between and .