Question

Sports car racing is a form of motorsport which uses sports car prototypes. The competition is held on special tracks designed in various shapes.

The equation of one such track is given as follows:

`f(x) = {{:((x^4-4x^2+4)",",0 ≤ x < 3), (x^2+40",",x≥3):}`

Based on given information, answer the following questions:

1. Find f'(x) for 0 < x < 3. 1
2. Find f'(4) 1
3. 1. Test for continuity of f(x) at x = 3. 2
      OR
   2. Test for differentiability of f(x) at x = 3. 2

Answer 1

Given,

 `f(x) = {{:((x^4-4x^2+4)",",0 ≤ x < 3), (x^2+40",",x≥3):}`

(i) f(x) = x⁴ – 4x² + 4 for 0 ≤ x < 3

Differential both sides w.r.t. ‘x’

f'(x) = 4x³ – 8x

(ii) f(x) = x² + 40, x ≥ 3

Differential both sides w.r.t. ‘x’

f'(x) = 2x

f'(x) = 2 × 4

f'(x) = 8

(iii) (a) LHL = RHL = f(3)

f(3) = 32 + 40 = 49

LHL = `lim_(x→3-)f(x)`

= `lim_(h→0)f(3-h)`

= `lim_(h→0)(3-h)^4-4(3-h)^2+4`

= `lim_(h→0)(3-h)^2(3-h)^2-4(3-h)^2+4`

= 81 – 36 + 4

= 49

RHL = `lim_(x→3^+)f(x)`

= `lim_(h→0)f(3+h)`

= `lim_(h→0)(3+h)^2+40`

= 9 + 40 

= 49

Hence, LHL = RHL = f(3) = 49

So, f(x) is continuous at x = 3

(iii) (b) `f(x) = {{:((x^4-4x^2+4)",",0 ≤ x < 3), (x^2+40",",x≥3):}`

`f'(x) = {{:((4x^3-8x)",",0 ≤ x < 3), (2x",",x>3):}`

For probability,

`x < 3: lim_(x->3^-)f'(x)=lim_(x->3^-)4(3)-8(3)`

= 108 – 24 

= 84

`x>3;lim_(x->3^+)f'(x)=lim_(x->3^+)`

= 2 × 3

= 6

So, Lf'(x) ≠ Rf'(x)

Hence, f(x) is not diff. at x = 3