Asked by Abigail
Identify the coordinates of any local extrema of the function y=e^x - e^(2x).
Answers
Answered by
MathMate
Let
f(x)=e<sup>x</sup> - e<sup>2x</sup>
the domain of f(x) is (-∞,∞).
Thus the extrema of f(x) can be found at point(s) where f'(x)=0.
f'(x)=e<sup>x</sup> - 2e<sup>2x</sup>
and f'(x)=0 when
e<sup>x</sup> = 2e<sup>2x</sup>
2e<sup>x</sup>=1
x=ln(1/2) (only root)
Since f"(x)=e<sup>x</sup> - 4e<sup>2x</sup>
and
f"(-ln(1/2)) = -1/2
we conclude that a maximum exists at x=ln(1/2) since f" is negative.
Can you take it from here?
f(x)=e<sup>x</sup> - e<sup>2x</sup>
the domain of f(x) is (-∞,∞).
Thus the extrema of f(x) can be found at point(s) where f'(x)=0.
f'(x)=e<sup>x</sup> - 2e<sup>2x</sup>
and f'(x)=0 when
e<sup>x</sup> = 2e<sup>2x</sup>
2e<sup>x</sup>=1
x=ln(1/2) (only root)
Since f"(x)=e<sup>x</sup> - 4e<sup>2x</sup>
and
f"(-ln(1/2)) = -1/2
we conclude that a maximum exists at x=ln(1/2) since f" is negative.
Can you take it from here?
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