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(C) Given equation is $f(x) = x^3+x-1 = 0$.To determine the number of real roots,we analyze the derivative of the function $f(x)$.The derivative is $f

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exactly one real root.

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(C) Given equation is $f(x) = x^3+x-1 = 0$.To determine the number of real roots,we analyze the derivative of the function $f(x)$.The derivative is $f'(x) = 3x^2 + 1$.Since $x^2 \geq 0$ for all $x \in \mathbb{R}$,it follows that $3x^2 + 1 \geq 1 > 0$ for all $x \in \mathbb{R}$.Because $f'(x) > 0$ for all $x$,the function $f(x)$ is strictly increasing on the entire real line.$A$ strictly increasing function can cross the $x$-axis at most once.We observe the values of the function at specific points:$f(0) = 0^3 + 0 - 1 = -1$$f(1) = 1^3 + 1 - 1 = 1$Since $f(0)  0$,by the Intermediate Value Theorem,there must exist at least one real root $x \in (0, 1)$ such that $f(x) = 0$.Since the function is strictly increasing,this root is unique.Therefore,the equation has exactly one real root.

The equation $x^3+x-1=0$ has

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