Question

Which of the following is correct about the function f(x)=|x^2+18x+81|?

(a) Left and right limits are equal and hence it is differentiable

(b) Limits are not important to determine differentiability

(c) Left and right limits are equal and hence it is not differentiable

(d) Left and right limits are not equal and hence it is not differentiable

I had been asked this question during a job interview.

This key question is from Differentiability in chapter Complex Function Theory of Engineering Mathematics

Answer 1

Correct option is (d) Left and right limits are not equal and hence it is not differentiable

To elaborate: f(x)=|(x+9)^2|

Since the break is at x=-9, we frame for the function for both the sides of -9.

f(x)=(x+9)^2, if x>-9 and f(x)=(x+9)^2, if x≤-9

We know that, a function is not differentiable at point x=a, if either \(\lim_{h \to 0}\frac{f(a+h)-f(a)}{h}\)does not exist or is infinity.  We check limits for both the cases of the function.

Left limit: Here, a=-10.

\(\lim_{h \to 0}\frac{f(-10+h)-f(-10)}{h}\)

\(\lim_{h \to 0}\frac{(-10+h+9)^2-(-10+9)^2}{h}\)

\(\lim_{h \to 0}\frac{1^2+h^2-2h-1^2}{h}\)

=-2

Right limit: Here, a=10.

\(\lim_{h \to 0}\frac{f(1+h)-f(1)}{h}\)

\(\lim_{h \to 0}\frac{(10+h+9)^2-(10+9)^2}{h}\)

\(\lim_{h \to 0}\frac{10^2+h^2+20h-10^2}{h}\)

=20

Since the two limits are not equal, the function is not differentiable.