Chapter 5.1, Problem 54E

a.

To determine

To show: that there is an open interval containing c such that $f\left(x\right)-f\left(c\right)\le 0$ for all x in the open interval.

Explanation of Solution

Given information:

The function has a local maximum value at the interior point c of its domain and that $f\text{'}\left(c\right)$ exists.

Proof: since, the function has a local maximum value at the interior point c of its domain and also $f\text{'}\left(c\right)$ exists,

Let open interval $\left(a,b\right)$ contains c , in this open interval for all x values the function value is always less than $f\left(c\right)$ ,

  $f\left(x\right)<f\left(c\right)$

So that for all x in the interval $f\left(x\right)-f\left(c\right)\le 0$ .

b.

To determine

To state: that why $\underset{x\to c^{+}}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}\le 0$ .

Answer to Problem 54E

when the value of x is tends to greater than c the value of function must be decrease, so the limit of at these values must be less than zero.

Explanation of Solution

Given information:

The function has a local maximum value at the interior point c of its domain and that $f\text{'}\left(c\right)$ exists.

The function has a local maximum value at the interior point c of its domain and that $f\text{'}\left(c\right)$ exists so when the value of x is tends to greater than c the value of function must be decrease, so the limit of at these values must be less than zero,

So we can say $\underset{x\to c^{+}}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}\le 0$ .

c.

To determine

To state: that why $\underset{x\to c^{-}}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}\ge 0$ .

Answer to Problem 54E

when the value of x is tends to less than c the value of function must be increase, so the limit of at these values must be greater than zero.

Explanation of Solution

Given information:

The function has a local maximum value at the interior point c of its domain and that $f\text{'}\left(c\right)$ exists.

The function has a local maximum value at the interior point c of its domain and that $f\text{'}\left(c\right)$ exists so when the value of x is tends to less than c the value of function must be increase, so the limit of at these values must be greater than zero,

So we can say $\underset{x\to c^{-}}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}\ge 0$ .

d.

To determine

To state: that how parts (b) and (c) allow us to conclude $f\text{'}\left(c\right)=0$ .

Answer to Problem 54E

The left and right limit exist.

Explanation of Solution

Given information:

The function has a local maximum value at the interior point c of its domain and that $f\text{'}\left(c\right)$ exists.

because the limit of function exist,

  $\underset{x\to c^{-}}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}\ge 0$ and $\underset{x\to c^{-}}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}\ge 0$ ,

So the derivative of function must be exist,

Thus, $f\text{'}\left(c\right)=0$ exists.

e.

To determine

To state: the argument if function has a local minimum value at an interior point.

Answer to Problem 54E

If function has a local minimum value at the interior point of its domain then the derivative of function on that exists.

Explanation of Solution

Given information:

The function has a local maximum value at the interior point c of its domain and that $f\text{'}\left(c\right)$ exists.

If function has a local minimum value at the interior point of its domain then the derivative of function on that exists.