Mathematics - Application of Derivative Question with Solution | TestHub

The fundamental theorem and chain rule give f'(x)=2x(x²−1)(x²−4)(x²−9)=2x⁷−28x⁵+98x³−72x. Its seven distinct zeros 0, ±1, ±2, ±3 are critical points, so D is false. Differentiating, f''(x)=14x⁶−140x⁴+294x²−72 and f'''(x)=28x(3x⁴−20x²+21). For x>0, f'''(x)=0 means u=x² is either (10−√37)/3 or (10+√37)/3. Both are positive, so exactly two positive roots exist: B is true and C false. Rolle's theorem between 0,1,2,3 gives three positive zeros of f''. It also gives three negative zeros; because f'' has degree six there are no others, so A is false. Only B is true.