If f (x) = |x| 3 , show that f″(x) exists for all real x and find it.

Question

Philoid · Accepted Answer

When, x ≥ 0,

f(x) = |x|³ = x³

So, f’(x) = 3x²

And f’’(x) = d(f’(x))/dx = 6x

∴ f’’(x) = 6x

When x < 0,

f(x) = |x|³ = ( – x)³ = – x³

f’(x) = – 3x²

f’’(x) = – 6x

∴

If f (x) = |x|3, show that f″(x) exists for all real x and find it.