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Problem 51 Hard Difficulty

If $ f(x) = x^2 + 10 \sin x $, show that there is a number $ c $ such that $ f(c) = 1000 $.

Answer

$f(x)=x^{2}+10 \sin x$ is continuous on the interval $[31,32], f(31) \approx 957,$ and $f(32) \approx 1030 .$ since $957<1000<1030$
there is a number $c$ in (31,32) such that $f(c)=1000$ by the Intermediate Value Theorem. Note: There is also a number $c$ in
(-32,-31) such that $f(c)=1000$

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Video Transcript

All right, we have a function F of x equals x squared plus 10 times sine of X. And we want to show that there is some numbers. See where uh FFC will equal 1000. And so I have this function graft on the desk. Most graphing calculator. Here it is. So here's the great for the function expert plus 10 times sine of X. Now, what I want you to notice is that okay? Well, here is a point on the graph. There's some X move that back up. There is uh some just looking worried. Okay, here here's your horizontal X axis right here. I don't care if I don't touch this move the graph here is your horizontal X axis right here. Now uh here is where it looks like The function would equal 1000. So there's some point right here. Okay, approximately here where the function equals 1000. So there is some corresponding X value down here on the X axis where the function equals 1000. But to be more formal uh more mathematically correct as to why? Uh there is a number, see down here on the X axis where the function does have a value of 1000. Uh This X square plus 10 times sine of X is a continuous function. Clearly for these x values here, The function is less than 1000. And clearly for X values here to function is greater than 1000. So because F of X is a continuous function and there are X values where the function is less than 1000 and X values. Where it's greater than 1000 then there must be some X value in between those other X values where the function is equal to exactly 1000. So because f of X is less than 1000, and f of x elsewhere is greater than 1000. And most importantly, because f of X is a continuous function, uh there will be a point a number see, on the X axis where the function at See will equal 1000.