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When is the function defined by $f(x)=x^2+e^{-2x}$ increasing? I know you have to take the derivative and use certain values of $x$, but I am confused on how to do this particular problem, and I would love it if any could help. Any help is appreciated thanks.

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A differentiable function (such as this one) is increasing when its derivative is positive (and decreasing when its derivative is negative). Can you figure out when the derivative of $f$ is positive? – 6005 Apr 30 '13 at 22:53
is the derivative 2x + e^-2x? – Captn Buzz Apr 30 '13 at 22:57
No it's not, see answer below. The minus two is there because we are taking the derivative of a composed function. – justt Apr 30 '13 at 23:02

First take the derivative:


Now consider the values when the derivative is equal to 0:


Now the function is increasing whenever $f'(x)$ is positive, and decreasing when it is negative. I.e., $f$ is increasing when: \begin{align*} 2x &> 2e^{-2x}\\ x &> e^{-2x}\\ \ln(x) &> -2x \end{align*}

As @CommonerG says below, you can compute a decimal approximation to this root.

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$\frac{\delta f(x)}{\delta x}=-2e^{-2x}+2x$

Thus, the function is increasing when:



A computational approach to solving this inequality provides the condition:


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