​ (i) Define |x| (a) |x| =√(x^2) (b) |x| = x (c) |x| = -x (d) |x| = x^2 ​

Question

(i) Define |x|
(a) |x| = \(\sqrt{x^2}\)
(b) |x| = x
(c) |x| = -x
(d) |x| = x²
(ii) At which point \(\frac{d} {dx}\)|x| does not exist?
Find \(\frac{d} {dx}\) |x|.
(iii) Find \(\frac{d} {dx}\)|x³ – 7x| . Also, find the point at which the derivative exists.

Answer 1

(i) (a) |x| = \(\sqrt{x^2}\)

(ii) At x = 0, \(\frac{d}{dx}\) |x| does not exist.

(iii) 

Does not exist at
x³ – 7x = 0 ⇒ x(x² – 7) = 0
⇒ x = 0, x² – 7 = 0 ⇒ x = ±7–√.