3. maximum mark: 6 find values for the constants a, b, and c that will make ( f(x)=sin x ) and ( g(x)=a +…

Explanation:

Step1: Find (f(0)), (g(0))

• (f(x)=\sin x), so (f(0)=\sin(0) = 0)
• (g(x)=a + bx+cx^{2}), so (g(0)=a)
• Since (f(0)=g(0)), then (a = 0)

Step2: Find (f^{\prime}(x)), (g^{\prime}(x)) and use (f^{\prime}(0)=g^{\prime}(0))

• (f^{\prime}(x)=\cos x), so (f^{\prime}(0)=\cos(0)=1)
• (g^{\prime}(x)=b + 2cx), so (g^{\prime}(0)=b)
• Since (f^{\prime}(0)=g^{\prime}(0)), then (b = 1)

Step3: Find (f^{\prime\prime}(x)), (g^{\prime\prime}(x)) and use (f^{\prime\prime}(0)=g^{\prime\prime}(0))

• (f^{\prime\prime}(x)=-\sin x), so (f^{\prime\prime}(0)=-\sin(0) = 0)
• (g^{\prime\prime}(x)=2c), so (g^{\prime\prime}(0)=2c)
• Since (f^{\prime\prime}(0)=g^{\prime\prime}(0)), then (2c=0), so (c = 0)

Answer:

(a = 0), (b = 1), (c = 0)