Question

Let f(x) = x | x | and g(x) = sin x

Statement I gof is differentiable at x = 0 and its derivative is continuous at that point.

Statement II gof is twice differentiable at x = 0.

Options

• Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I.

• Statement I is true, Statement II is true; Statement II is not a correct explanation of Statement I.

• Statement I is true, Statement II is false.

• Statement I is false, Statement II is true.

Answer 1

Statement I is true, Statement II is false.

Explanation:

Given, f(x) = x |x| and g(x) = sin x

gof (x) = sin (x |x|) = `{{:(-sinx^2",", x < 0),(sinx^2",", x ≥ 0):}`

(gof)'(x) = `{{:(-2x  cosx^2",", x < 0),(2x  cos x^2",", x ≥ 0):}` 

Clearly, L (gof)'(0) = 0 = R (gof)'(0)

∴ got is differentiable at x = 0 and also its derivative is continuous at x = 0.

 Now, (gof)''(x) = `{{:(-2 cos x^2  +  4x^2 sinx^2",", x < 0),(2 cos x^2  -  4x^2 sinx^2",", x > 0):}`

∴ L (gof)''(0) = – 2 and R (gof)''(0) = 2

∴ L (gof)''(0) ≠ R (gof)''(0)

∴ gof (x) is not twice differentiable at x = 0.