MHB True or False Integral Calculus Question #3

[MermaidWonders]

True or False: If $f(x)$ is a negative function that satisfies $f'(x) > 0$ for $0 \le x \le 1$, then the right hand sums always yield an underestimate of $\int_{0}^{1} (f(x))^2\,dx$.

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Would it be true since right hand Riemann sums for a negative, increasing function will always produce an underestimate for the integral, so it doesn't really matter if the entire "function" we're dealing with is squared?

 

[Greg]

$(f(x))^2\ge0$

 

[MermaidWonders]

Or... would it be that squaring $f(x)$ will turn $f(x)$ into a positive function from a negative function, so the statement is going to be false because taking the right Riemann sum will still give you an overestimate for positive, increasing functions?

 

[MermaidWonders]

Yeah, OK. So, in that case, it becomes a positive, increasing function, so it's an overestimate, right?

 

[Greg]

That's right.