Question

Evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result. (Round your answer to three decimal places.)f(x) = 3x + 9 (0, 1)f ''(0) =

Answer 1

The second derivative f''(x) of the function f(x) = 3x + 9 is 0. This result is constant for all values of x since the first derivative f'(x) is a constant 3. Verifying with a computer algebra system will confirm that f''(0) is 0.

Step-by-step explanation:

The initial function given is f(x) = 3x + 9. To find the second derivative, f''(x), we first need to calculate the first derivative, f'(x). The first derivative of a linear function is simply the coefficient of x, which in this case is 3. Therefore, f'(x) = 3.

Since the first derivative is a constant, the second derivative, which represents the rate of change of the first derivative with respect to x, will be zero. Thus, f''(x) = 0, regardless of the value of x. When evaluating the second derivative at the point (0, 1), we confirm that f''(0) = 0.

To verify this result using a computer algebra system, one would enter the original function and use the system's differentiation tool to calculate the second derivative. The output should confirm that the second derivative is indeed zero.

Answer 2

We want to evaluate the second derivative of the following function

`f(x)=(3)/(√(x+9))`

at x = 0.

To calculate the derivative of this function, we just have to use the power rule and along with the chain rule. Those rules are

`\begin{gathered} (d)/(dx)x^n=nx^(n-1) \\ (f(g(x)))^(\prime)=f^(\prime)(g(x))g^(\prime)(x) \end{gathered}`

Then, we have

`\begin{gathered} f(x)=(3)/(√(x+9))=3(x+9)^(-1/2) \\ f^(\prime)(x)=3\cdot(-(1)/(2))(x+9)^(-1/2-1)\cdot(x+9)^(\prime)=(3)/(2)(x+9)^(-3/2) \\ f^(\prime\prime)(x)=-(3)/(2)\cdot(-(3)/(2))(x+9)^(-3/2-1)\cdot(x+9)^(\prime)=-(9)/(4)(x+9)^(-5/2) \\ f^{^{\prime\prime^(\prime)}}(x)=(9)/(4)(1)/(√((x+9)^5)) \end{gathered}`

Now that we have the function, we just have to evaluate the function at x = 0.

`f^(\prime\prime)(0)=(9)/(4)(1)/(√((0+9)^5))=(9)/(4)(1)/(9^2√(9))=(9)/(4\cdot9^2\cdot3)=(1)/(108)\approx0.009`

TThe second derivative of this function evaluated at x = 0 is 0.009.