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 

(i) Find \(f'(x)\) for \(0<x>3\). 
(ii) Find \(f'(4)\). 
(iii)(a) Test for continuity of \(f(x)\) at \(x=3\). 
OR 
(iii)(b) Test for differentiability of \(f(x)\) at \(x=3\). 
 

Hint:

For piecewise functions, always check continuity first using LHL, RHL and function value. If the function is continuous, then compare left and right derivatives to test differentiability.

Answer 1

Step 1: Differentiate \(f(x)\) for \(0<x<3\).
For the interval \(0<x<3\): \[ f(x)=x^4-4x^2+4 \] Differentiate using the power rule: \[ f'(x)=4x^3-8x \] Thus \[ f'(x)=4x^3-8x \quad \text{for } 0<x<3 \] Step 2: Find \(f'(4)\).
For \(x \ge 3\): \[ f(x)=x^2+40 \] Differentiate: \[ f'(x)=2x \] Thus \[ f'(4)=2(4)=8 \] Step 3: Test continuity at \(x=3\).
A function is continuous at \(x=a\) if \[ \lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=f(a) \] Left hand limit: \[ \lim_{x\to3^-}(x^4-4x^2+4) \] Substitute \(x=3\): \[ 3^4-4(3^2)+4 \] \[ 81-36+4 \] \[ =49 \] Right hand limit: \[ \lim_{x\to3^+}(x^2+40) \] \[ =9+40 \] \[ =49 \] Function value: \[ f(3)=3^2+40=49 \] Since \[ \text{LHL}=\text{RHL}=f(3) \] the function is continuous at \(x=3\).
Step 4: Test differentiability at \(x=3\).
Compute left derivative. For \(0<x<3\): \[ f'(x)=4x^3-8x \] Thus \[ f'_-(3)=4(27)-8(3) \] \[ =108-24 \] \[ =84 \] Now compute right derivative. For \(x\ge3\): \[ f'(x)=2x \] Thus \[ f'_+(3)=2(3)=6 \] Since \[ f'_-(3)\ne f'_+(3) \] the function is not differentiable at \(x=3\).
Final Answer:
\[ f'(x)=4x^3-8x \quad (0<x<3) \] \[ f'(4)=8 \] The function is continuous at \(x=3\) but not differentiable at \(x=3\).