A calculus problem by Ankit Nigam

Calculus Level 3

Let $h\left( x \right) =f\left( x \right) -{ (f\left( x \right) ) }^{ 2 }+{ (f\left( x \right) ) }^{ 3 }$ for every real number x. Then

PRACTICE FOR BITSAT HERE

h is increasing whenever f is increasing nothing can be said in general h is increasing whenever f is decreasing h is decreasing whenever f is increasing

1 solution

Aalap Shah
Apr 17, 2015

Differentiate both sides of the given equation, with respect to x , using chain rule: $\therefore h'(x)=f'(x)-2f(x)f'(x)+3{ f(x) }^{ 2 }f'(x)$ $\therefore h'(x)=f'(x)[1-2f(x)+3{ f(x) }^{ 2 }]$ Note that the quantity in the bracket is always positive, provided f(x) is real. This is because the quadratic discriminant of: $3x^2-2x+1$ is less than zero. Thus, h'(x) and f'(x) have the same sign, implying that h(x) is increasing iff f(x) is increasing.

A calculus problem by Ankit Nigam

1 solution

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