Question

1. let f be a differentiable function such that f(3)=4 and f(3)=5. if g(x)=x²f(x), what is the value of g(3)?

Explanation:

Step1: Apply the product rule

The product rule states that if \(g(x)=u(x)v(x)\), then \(g'(x)=u'(x)v(x)+u(x)v'(x)\). Here, \(u(x) = x^{2}\) and \(v(x)=f(x)\). So, \(u'(x) = 2x\) and \(v'(x)=f'(x)\). Then \(g'(x)=2x\cdot f(x)+x^{2}\cdot f'(x)\).

Step2: Substitute \(x = 3\)

We know that \(f(3) = 4\) and \(f'(3)=5\). Substitute \(x = 3\) into \(g'(x)\):

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Answer:

D. 69