Answer to Question #165225 in Calculus for Johnny Tim

Question

Let f be the function given by f (x) = |x|. Which of the following statements about f is true?

I. f is continuous at x = 0

II. f is differentiable at x = 0

III. f has an absolute maximum at x = 0.

Answer 1

Ans:-

f(x) $\Rightarrow$ $\begin{cases} x&x\ge0\\-x&x<0\end{cases}$

At $x<0$ , f(x)=-x is continuous every where

At $x>0$ , f(x)=x is continuous every where

at x=0

L.H.L.

$lim_{x \to o^-} f(x)=$ $lim_{x \to o} -x=$ 0

R.H.L.

$lim_{x \to o^+} f(x)=$ $lim_{x \to 0} x=$0

f(0)=0

L.H.L=R.H.L=f(0)

f(x) is continuous at x=0

Now,

f'(x)$\Rightarrow$ $\begin{cases} -1&x<0\\1&x\ge0\end{cases}$

At x=0, f'(x) is not differentiable

f''(x)=0 then

So f(x) is neither maximum nor minimum at x=0

Hence Given statement is false and first option is true which is f(x) is continuous at x=0