Question

graph the cube root function and analyze the minimum and maximum on the given interval. ( h(x) = sqrt3{4x - 5}, -8,4 ) choose the correct graph below. (options a, b, c, d with graphs)

Explanation:

Step1: Analyze the function \( h(x) = 4\sqrt[3]{x - 5} \)

The parent function is \( y=\sqrt[3]{x} \), which has a domain of all real numbers and is increasing everywhere. The transformation here is a horizontal shift right by 5 units and a vertical stretch by a factor of 4. So \( h(x) \) is also an increasing function (since the cube root function is increasing and stretching vertically doesn't change the monotonicity).

Step2: Evaluate at endpoints of \([-8, 4]\)

• At \( x=-8 \): \( h(-8)=4\sqrt[3]{-8 - 5}=4\sqrt[3]{-13}\approx4\times(-2.351)= - 9.404 \)
• At \( x = 4 \): \( h(4)=4\sqrt[3]{4 - 5}=4\sqrt[3]{-1}=4\times(-1)=-4 \)

Since the function is increasing on \(\mathbb{R}\), on the interval \([-8,4]\), the minimum is at \( x=-8 \) and the maximum is at \( x = 4 \). Now, we analyze the graph's behavior: as \( x \) increases, \( h(x) \) increases. Let's check the graphs:

• Graph A: Let's see the trend. If we check the key points, when \( x = 5 \), \( h(5)=4\sqrt[3]{0}=0 \). For \( x < 5 \), \( x - 5<0 \), so \( h(x)<0 \); for \( x>5 \), \( h(x)>0 \). In the interval \([-8,4]\), all \( x \) are less than 5, so \( h(x)<0 \). Now, as \( x \) goes from -8 to 4 (increasing), \( h(x) \) should increase from \( h(-8)\approx - 9.4 \) to \( h(4)=-4 \). So the graph should be increasing (going up as \( x \) increases) in the third quadrant (since \( x \) negative, \( y \) negative) moving towards \( x = 4 \), \( y=-4 \).

Looking at the options, we need to find the graph that is increasing (since the function is increasing) on \([-8,4]\) and has the correct \( y \)-values at the endpoints.

After analyzing the function's behavior (increasing, \( h(-8)\approx - 9.4 \), \( h(4)=-4 \)), the correct graph should show an increasing curve from a lower \( y \)-value (around -9.4) at \( x=-8 \) to -4 at \( x = 4 \). Among the options, we can identify the correct graph (assuming the visual cues match the function's properties).

Answer:

(Assuming the correct graph is, for example, Option A after visual verification based on the function's properties. Since the exact visual is needed, but based on the function analysis, the correct graph should be the one that is increasing on \([-8,4]\) with \( h(-8)\approx - 9.4 \) and \( h(4)=-4 \). If we assume the correct option is A, then) A. (Graph corresponding to the increasing cube root function with the correct endpoint values)