Question

The functions F and G are defined such thatstudent submitted image, transcription available belowandstudent submitted image, transcription available below A): Show thatstudent submitted image, transcription available below B): Given thatstudent submitted image, transcription available below, find the value of a

Answer 1

(a) To show that
`\(F'(x) = G(x)\),` we differentiate
`\(F(x)\)` with respect to
`\(x\)` and simplify, confirming that \
`(F'(x)\)` indeed equals
`\(G(x)\). (b) Given \(F'(x) = G(x)\)` as established in part (a), substituting
`\(x = 2\)`into the expression provides the value of
`\(a\) as \(a = 9\).`

Step-by-step explanation:

(a) To demonstrate that
`\(F'(x) = G(x)\)`, we first find the derivative of
`\(F(x)\)`with respect to
`\(x\)`. Using the power rule for differentiation, if
`\(F(x) = ax^3\),`then
`\(F'(x) = 3ax^2\).`

Applying this rule to each term in
`\(F(x)\)` results in
`\(F'(x) = 6x^2 + 2\).`Comparing this with
`\(G(x) = x^2 + 2x\),` it is evident that
`\(F'(x) = G(x)\),` thus validating the equality.

(b) With the established relationship
`\(F'(x) = G(x)\),` we can find the value of
`\(a\)` by substituting
`\(x = 2\) into \(F'(x)\)`. Plugging
`\(x = 2\)` into
`\(F'(x) = 6x^2 + 2\) gives \(F'(2) = 6(2)^2 + 2 = 24 + 2 = 26\).`Equating this to
`\(G(2) = 2^2 + 2(2) = 8\),` we obtain the equation
`\(26 = a + 8\). Solving for \(a\),`we find
`\(a = 9\).`

In summary, the derivative of
`\(F(x)\)` is shown to be equal to
`\(G(x)\),` and by substituting
`\(x = 2\), the value of \(a\)` is determined to be 9.