Questions: By inspecting the graph of the function, find the absolute maximum and absolute minimum on the given interval. m(x)=x^2+1 on [-4,2] Find the absolute maximum. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The absolute maximum is at x= . (Use a comma to separate answers as needed.) B. There is no absolute maximum.

Solution Steps

To find the absolute maximum of the function \( m(x) = x^2 + 1 \) on the interval \([-4, 2]\), we need to evaluate the function at the critical points and the endpoints of the interval. The critical points are found by setting the derivative of the function to zero. After finding the critical points, we compare the function values at these points and the endpoints to determine the absolute maximum.

To find the critical points of the function \( m(x) = x^2 + 1 \), we first compute its derivative: \[ m'(x) = 2x \]

Setting the derivative equal to zero gives us: \[ 2x = 0 \implies x = 0 \] Thus, the critical point is \( x = 0 \).

Next, we evaluate the function at the critical point and the endpoints of the interval \([-4, 2]\):

• At \( x = 0 \): \[ m(0) = 0^2 + 1 = 1 \]
• At \( x = -4 \): \[ m(-4) = (-4)^2 + 1 = 16 + 1 = 17 \]
• At \( x = 2 \): \[ m(2) = 2^2 + 1 = 4 + 1 = 5 \]

Now we compare the values obtained:

• \( m(0) = 1 \)
• \( m(-4) = 17 \)
• \( m(2) = 5 \)

The maximum value is \( 17 \) at \( x = -4 \).

Final Answer