Select the true statements. \( \square \) A. The function \( f(x)=x^{3} \) is differentiable on \( [0,7] \). \( \square \) B. The function \( f(x)=\sqrt{x} \) is differentiable on \( [0,7] \) \( \square \) C. The function \( f(x)=|x| \) is differentiable on \( (-3,3) \) \( \square \) D. The function \( f(x)=\sqrt{x} \) is differentiable on \( (0,7) \) \( \square \) E. The function \( f(x)=|x| \) is differentiable on \( [0,7] \).

1. For statement A, the function is \( f(x)=x^3 \). - Since polynomials are differentiable everywhere, \( f(x) \) is differentiable on \([0,7]\). - Statement A is true. 2. For statement B, the function is \( f(x)=\sqrt{x} \). - Its derivative is \( f'(x)=\frac{1}{2\sqrt{x}} \). - At \( x=0 \), the derivative would require evaluating \(\frac{1}{2\sqrt{0}}\), which is undefined (or infinite). - Thus, \( f(x)=\sqrt{x} \) is not differentiable at \( x=0 \), and consequently not differentiable on the entire interval \([0,7]\). - Statement B is false. 3. For statement C, the function is \( f(x)=|x| \). - The function \( |x| \) has a derivative of \(-1\) for \( x<0 \) and \( 1 \) for \( x>0 \). - At \( x=0 \), the left-hand derivative is \(-1\) and the right-hand derivative is \(1\); they are not equal. - Hence, \( f(x) \) is not differentiable at \( x=0 \), so it is not differentiable on the entire interval \((-3,3)\). - Statement C is false. 4. For statement D, the function is again \( f(x)=\sqrt{x} \), but now the interval is \((0,7)\). - On \((0,7)\), all \( x \) are positive, so the derivative \( f'(x)=\frac{1}{2\sqrt{x}} \) exists for every \( x \) in the interval. - Therefore, \( f(x)=\sqrt{x} \) is differentiable on \((0,7)\). - Statement D is true. 5. For statement E, the function is \( f(x)=|x| \) on \([0,7]\). - For \( x>0 \), \( f(x)=|x|=x \) which is differentiable with derivative \( 1 \). - At the endpoint \( x=0 \) (since this is a closed interval), we only need to check the right-hand derivative. - The right-hand derivative at \( x=0 \) is \[ \lim_{h \to 0^+} \frac{|h|-0}{h} = \lim_{h \to 0^+} \frac{h}{h} = 1. \] - Since the one-sided derivative exists at \( x=0 \) and the function is differentiable for all \( x>0 \), \( f(x)=|x| \) is differentiable on \([0,7]\). - Statement E is true. The true statements are A, D, and E.