Question

1. let f and g be differentiable functions such that f(1) = 2 and g(1) = 6. if h(x) = 5f(x) - 4g(x) + 3x² - 2, what is the value of h(1)?

Explanation:

Step1: Differentiate \( h(x) \)

Using the sum and constant - multiple rules of differentiation. If \( h(x)=5f(x)-4g(x)+3x^{2}-2 \), then \( h^{\prime}(x)=5f^{\prime}(x)-4g^{\prime}(x)+6x \).

Step2: Substitute \( x = 1 \)

We know that \( f^{\prime}(1)=2 \) and \( g^{\prime}(1)=6 \). Substitute these values into \( h^{\prime}(x) \):
\( h^{\prime}(1)=5\times f^{\prime}(1)-4\times g^{\prime}(1)+6\times1 \)
\( h^{\prime}(1)=5\times2-4\times6 + 6\)
\( h^{\prime}(1)=10-24 + 6\)
\( h^{\prime}(1)=-8\)

Answer:

B. - 8