23. 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)?
23. 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)?
Answer
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)): [ \begin{align*} g'(3)&=2\times3\times f(3)+3^{2}\times f'(3)\ &=6\times4 + 9\times5\ &=24+45\ &=69 \end{align*} ]
Answer:
D. 69