If f(x) = x^{3} and g(x) = x^{3} - 4x in the | vedclass

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(B) Given $f(x) = x^{3}$ and $g(x) = x^{3} - 4x$ on $[-2, 2]$.Since $f(x)$ and $g(x)$ are polynomials,they are continuous on $[-2, 2]$ and differentia

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$(a)$ and $(c)$ are correct

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(B) Given $f(x) = x^{3}$ and $g(x) = x^{3} - 4x$ on $[-2, 2]$.Since $f(x)$ and $g(x)$ are polynomials,they are continuous on $[-2, 2]$ and differentiable on $(-2, 2)$. Thus,both satisfy the Mean Value Theorem.For Rolle's Theorem,we check $f(a) = f(b)$:$f(-2) = (-2)^{3} = -8$ and $f(2) = (2)^{3} = 8$. Since $f(-2) 
eq f(2)$,$f(x)$ does not satisfy Rolle's Theorem.$g(-2) = (-2)^{3} - 4(-2) = -8 + 8 = 0$ and $g(2) = (2)^{3} - 4(2) = 8 - 8 = 0$. Since $g(-2) = g(2)$,$g(x)$ satisfies Rolle's Theorem.Therefore,statement $(a)$ and statement $(c)$ are correct.

If $f(x) = x^{3}$ and $g(x) = x^{3} - 4x$ in the interval $[-2, 2]$,consider the following statements:
$(a)$ $f(x)$ and $g(x)$ satisfy the Mean Value Theorem.
$(b)$ $f(x)$ and $g(x)$ both satisfy Rolle's Theorem.
$(c)$ Only $g(x)$ satisfies Rolle's Theorem.
Which of these statements is correct?

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If $f(x) = x^{3}$ and $g(x) = x^{3} - 4x$ in the interval $[-2, 2]$,consider the following statements:$(a)$ $f(x)$ and $g(x)$ satisfy the Mean Value Theorem.$(b)$ $f(x)$ and $g(x)$ both satisfy Rolle's Theorem.$(c)$ Only $g(x)$ satisfies Rolle's Theorem.Which of these statements is correct?