The numbers of the roots of $e^x + 0.5x^2 -2 = 0$ in the range $[-5, 5]$ are ________ .
My try:
Someone explained using intermediate value theorem, and someone used graph to show that this equation has two roots in given range.
But, this was an exam question. If someone does not know how to draw graph for this equation, then can we solve using any alternative method in easy way?
Can you please explain ?
$\endgroup$2 Answers
$\begingroup$Let $f(x):=e^x + 0.5x^2 -2$. We first note that $f(-5)>0$, $f(0)<0$, and $f(5)>0$ which imply, by continuity, that there are at least two roots in the interval $[-5,5]$: at least one in $(-5,0)$, and at least one in $(0,5)$.
Moreover $f$ is strictly convex because the second derivative $f''(x)=e^x+1$ is strictly positive. Note that if $f''>0$ then $f'$ is strictly increasing and therefore $f'$ can have at most one zero. By then MVT this implies that $f$ has at most two roots (in any interval).
Then we conclude that $f$ has exactly two roots in the interval $[-5,5]$.
$\endgroup$ 5 $\begingroup$Rewrite the equation as $e^x=2-{1\over2}x^2$ and draw two graphs: the exponential curve $y=e^x$ and the parabola $y=2-{1\over2}x^2$. Even a rough sketch should tell you these two curves have two points of intersection.
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