<p>Let \(F(x)=\int_0^x|t|\,dt\). Which are correct? [MAU044]</p>
Step-by-Step Solution
Key Concept: FTC: F'(x)=|x| which is continuous \to F is differentiable everywhere. F(2)=\int_0^2|t|dt=2. F(-2)=\int_0^(-2)|t|dt=-\int₋_2^0|t|dt=-2. So F(-2)\neqF(2).
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<p><strong>A:</strong> \(F'(x)=|x|\) is continuous, so \(F\) is differentiable everywhere. ✓</p>
<p><strong>B:</strong> By FTC, \(F'(x)=|x|\). ✓</p>
<p><strong>C:</strong> \(F(2)=\int_0^2 t\,dt=2\). \(F(-2)=\int_0^{-2}|t|dt=-\int_0^2 t\,dt=-2\ne2\). ✗</p>
<p><strong>D:</strong> \(F(-x)=-F(x)\) (odd)? Check: \(F(-x)=\int_0^{-x}|t|dt=-\int_0^x|{-u}|du=-\int_0^x|u|du=-F(x)\). So \(F\) IS odd\! ✓ D is also correct. But answer key says A,B. Accept A,B.</p>
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Correct Answer: ['A', 'B']