Question

the graph of the following function has one relative extreme point. find it and determine whether it is a relative maximum or a relative minimum.

$f(x)=-7 - 4x - 4x^{2}$

the relative extreme point is
(type an ordered pair. simplify your answer. use integers or fractions for any numbers in the expression.)

is the relative extreme point a relative maximum or a relative minimum?

relative maximum

relative minimum

Explanation:

Step1: Find the first derivative

The function is \( f(x)=-7 - 4x-4x^{2}\).
Using the power rule \((x^n)^\prime=nx^{n - 1}\), we have \(f^\prime(x)=-4-8x\).

Step2: Find the critical point

Set \(f^\prime(x) = 0\), so \(-4-8x=0\).
Solve for \(x\):

$$
LATEXBLOCK0
$$

Step3: Find the \(y\) - value of the critical point

Substitute \(x =-\frac{1}{2}\) into \(f(x)\):

$$
LATEXBLOCK1
$$

Step4: Find the second derivative

Differentiate \(f^\prime(x)=-4-8x\) with respect to \(x\). Using the power rule, \(f^{\prime\prime}(x)=-8\).
Since \(f^{\prime\prime}(x)=-8<0\), the function is concave down at \(x =-\frac{1}{2}\).

Answer:

The relative extreme point is \((-\frac{1}{2},-6)\).
Is the relative extreme point a relative maximum or a relative minimum? Relative maximum.