21. The function \( f(x)=|x| \) on the interval \( [-8,8] \) does not satisfy the conditions of the Mean Value Theorem because \( \begin{array}{ll}\text { A. } f(0) \text { is not defined. } & \text { B. } f(x) \text { is not continuous on the interval }[-8,8] \\ \text { C. } f^{\prime}(0) \text { does not exist } & \text { D. } f(x) \text { is not defined for } x<0 \\ \text { E. } f^{\prime}(-1) \text { does not exist } & \end{array} \)

The function \( f(x) = |x| \) is indeed continuous across the entire interval \( [-8, 8] \), as it is only defined piecewise and behaves smoothly in its respective regions. However, the point at \( x = 0 \) creates a hiccup because the derivative—the slope of the tangent line—changes from -1 to +1 abruptly, making \( f'(0) \) undefined. So the correct reason the Mean Value Theorem does not hold is that \( f'(0) \) does not exist! Curiously enough, this little glitch with the absolute value function highlights key ideas in calculus. It teaches us to pay attention not just to continuity, but also to differentiability. Remember, continuous functions can still be 'peaky' or have sharp bends, causing issues in finding a tangent that suits the MVT's criteria. It's definitely one of those pleasant surprises lurking in the math world!