Đề bài

Nếu \(f(x) = {\sin ^2}x + x{e^{2x}}\) thì \(f''(0)\) bằng

A. 4

B. 5

C. 6

D. 0

Lời giải chi tiết

\(f'(x) = 2\sin x.(\sin x)' + x'.{e^{2x}} + x.({e^{2x}})'\) \( = 2\sin x.\cos x + {e^{2x}} + x.(2x)'{e^{2x}}\) \( = 2\sin x.\cos x + {e^{2x}} + 2x{e^{2x}}\) \( = \sin 2x + {e^{2x}}(2x + 1)\). \(f''(x) = (2x)'\cos 2x + ({e^{2x}})'(2x + 1) + {e^{2x}}(2x + 1)'\) \( = 2\cos 2x + (2x)'{e^{2x}}(2x + 1) + {e^{2x}}.2\) \( = 2\cos 2x + 2{e^{2x}}(2x + 1) + 2{e^{2x}}\) \( = 2\cos 2x + 4x{e^{2x}} + 2{e^{2x}} + 2{e^{2x}}\) \( = 2\cos 2x + 4x{e^{2x}} + 4{e^{2x}}\). \(f''(0) = 2\cos 2.0 + 4.0{e^{2.0}} + 4{e^{2.0}} = 2 + 0 + 4 = 6\).

Đáp án C