Identify the function's local and absolute extreme values, if any, saying where they occur.
-$$g ( x ) = \frac { x ^ { 4 } } { 4 } - \frac { 7 } { 3 } x ^ { 3 } + 7 x ^ { 2 } - 8 x - 4$$
A) local maxima at x = 1 and x = 4; local minimum at x = 2
B) local maxima at x = -1 and x = -4; local minimum at x = 2
C) local maximum at x = 1; local minimum at x = 4
D) local maximum at x = 2; local minima at x = 1 and x = 4

Question

Quizplus · Accepted Answer

The function is a quartic polynomial, so there are no vertical asymptotes or holes in the graph. To determine the local and absolute extreme values, we can take the derivative of the function and set it equal to zero to find critical points.

$g'(x) = x^3-7x^2+14x-8$

Setting $g'(x) = 0$, we get

$(x-1)(x-2)(x-4) = 0$

So the critical points are $x = 1$, $x = 2$, and $x = 4$. To determine whether these are local maxima, local minima, or neither, we can use the second derivative test.

$g''(x) = 3x^2-14x+14$

At $x = 1$, $g''(1) = 3-14+14 = 3 > 0$, so there is a local minimum at $x = 1$.

At $x = 2$, $g''(2) = 12-28+14 = -2 < 0$, so there is a local maximum at $x = 2$.

At $x = 4$, $g''(4) = 48-56+14 = 6 > 0$, so there is a local minimum at $x = 4$.

Since there is no global maximum or minimum for this function (as $g(x)$ grows large in both negative and positive directions), the only extreme values occur at the critical points we found: a local maximum at $x = 2$ and local minima at $x = 1$ and $x = 4$. Therefore, the best choice is D.

Identify the Function's Local and Absolute Extreme Values, If Any g(x)=x44−73x3+7x2−8x−4g ( x ) = \frac { x ^ { 4 } } { 4 } - \frac { 7 } { 3 } x ^ { 3 } + 7 x ^ { 2 } - 8 x - 4g(x)=4x4​−37​x3+7x2−8x−4

Identify the Function's Local and Absolute Extreme Values, If Any $g ( x ) = \frac { x ^ { 4 } } { 4 } - \frac { 7 } { 3 } x ^ { 3 } + 7 x ^ { 2 } - 8 x - 4$