# What are the inflections points of #y= e^(2x) - e^x #?

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To find the inflection points of ( y = e^{2x} - e^x ), we first find the second derivative and set it equal to zero to find the points of inflection.

First derivative: [ \frac{dy}{dx} = 2e^{2x} - e^x ]

Second derivative: [ \frac{d^2y}{dx^2} = 4e^{2x} - e^x ]

Setting the second derivative equal to zero: [ 4e^{2x} - e^x = 0 ]

Dividing both sides by ( e^x ): [ 4e^x - 1 = 0 ]

Adding 1 to both sides: [ 4e^x = 1 ]

Dividing both sides by 4: [ e^x = \frac{1}{4} ]

Taking the natural logarithm of both sides: [ x = \ln\left(\frac{1}{4}\right) ]

Solving for ( x ): [ x = -\ln(4) ]

Thus, the inflection point is ( x = -\ln(4) ).

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