SOLUTION 10: We are given the equation $ x^3 \cos x = 4 $, so let function

$$ f(x)= x^3 \cos x \ \ \ \ and \ choose \ \ \ \ m=4 $$

This function is continuous for all values of $x$ since it is the PRODUCT of continuous functions. We have the continuous polynomial function $x^3$, and $\cos x$ is a well-known continuous function. We now need to search for an appropriate interval satisfying the assumptions of the Intermediate Value Theorem. By trial and error, we have that

$$ f(0)= (0)^3 \cos (0) = (0)(1)= 0 < 4 \ \ \ \ and \ \ \ \ f(2\pi)= (2\pi)^3 \cos (2\pi) = 8\pi(1)=8\pi \approx 25.1 > 4 $$

so that $$ f(0)=0 < m < 8\pi = f(2\pi) $$

i.e., $m=4$ is between $ f(0) $ and $ f(2\pi) $.

The assumptions of the Intermediate Value Theorem have now been met, so we can conclude that there is some number $c$ in the interval $[0, 2\pi]$ which satisfies $$ f(c)=m $$ i.e., $$ c^3 \cos c = 4 $$ and the equation is solvable.

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