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Have the following

$f(x)=x^2 \exp(\sin(x))-\cos(x)$ on the interval $[0,\pi/2]$, I have shown the function is continuous and that there is at least one solution on the interval via using IVT, I know I have to find another solution in the interval such that $f(x)<0$, $f(x)>0$ and then $f(x)<0$, just wondering what is the best way to approach, Im sure there must be a more effective way than just to number crunch, could we consider MVT on the interval, many thanks in advance.

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    $\begingroup$ By "one solution," presumably you mean one solution to $f(x)=0$? Pedantic, yes, but a function is not an equation. $\endgroup$ May 1, 2012 at 15:29
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    $\begingroup$ Take the derivative. You will get a positive impression. $\endgroup$ May 1, 2012 at 15:30
  • $\begingroup$ Why do you think there is more than one solution to $f(x)=0$? $\endgroup$ May 1, 2012 at 15:31
  • $\begingroup$ @AndréNicolas many thanks for the input. $\endgroup$
    – user24930
    May 1, 2012 at 15:42
  • $\begingroup$ The answer of @Simon Markett is much better than my comment, and I am not referring to my feeble pun. Sure, if you find the derivative, it is obvious that it is positive, end of story. But looking directly at the two functions is the "right" approach, derivative is mechanical. Look and then (if necessary) compute is much better than compute and then look. $\endgroup$ May 1, 2012 at 15:48

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The function $x^2exp(sin(x))$ is increasing, the function $cos(x)$ is decreasing in your interval. Hence the combined functions is increasing on your interval. It starts with a value of $-1$ and goes up to $(\pi/2)^2e$. Therefore it has precisely one zero point.

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