Find all relative extrema of the function. (If an answer does not exist, enter DNE:) f(x) = 3x – 36x^(1/3) relative maximum (x, Y) = relative minimum (x, Y)
The Correct Answer and Explanation is:
To find the relative extrema of the function
f(x) = 3x – 36x^(1/3)
Step 1: Find the first derivative
We use the power rule:
f(x) = 3x – 36x^(1/3)
f′(x) = d/dx[3x] – d/dx[36x^(1/3)]
f′(x) = 3 – 36 * (1/3)x^(-2/3)
f′(x) = 3 – 12x^(-2/3)
Step 2: Set the derivative equal to zero
We solve for critical points by setting f′(x) = 0:
3 – 12x^(-2/3) = 0
3 = 12x^(-2/3)
Divide both sides by 12:
1/4 = x^(-2/3)
Now solve for x:
x^(-2/3) = 1/4
Take the reciprocal:
x^(2/3) = 4
Raise both sides to the 3/2 power:
x = ±(4)^(3/2) = ±8
So we get two critical points: x = 8 and x = -8
Step 3: Use the second derivative test or first derivative sign chart
Second derivative:
f′(x) = 3 – 12x^(-2/3)
Differentiate again:
f″(x) = d/dx[-12x^(-2/3)] = -12 * (-2/3)x^(-5/3) = 8x^(-5/3)
Now evaluate at x = 8:
f″(8) = 8*(8)^(-5/3) = 8 / (8^(5/3))
= 8 / [2^5] = 8 / 32 = 1/4 > 0
So f has a relative minimum at x = 8
Evaluate at x = -8:
f″(-8) = 8 * (-8)^(-5/3)
Negative base to an odd power gives negative result:
f″(-8) < 0
So f has a relative maximum at x = -8
Step 4: Find the y-values
f(8) = 3(8) – 36(8)^(1/3) = 24 – 36(2) = 24 – 72 = -48
f(-8) = 3(-8) – 36(-8)^(1/3) = -24 – 36(-2) = -24 + 72 = 48
Final Answer:
Relative maximum: (x, y) = (-8, 48)
Relative minimum: (x, y) = (8, -48)
Explanation
To find the relative extrema of the function f(x) = 3x – 36x^(1/3), we begin by taking the derivative to identify critical points. The derivative f′(x) = 3 – 12x^(-2/3) represents the slope of the tangent line. We set the derivative equal to zero to locate any stationary points. Solving the equation 3 – 12x^(-2/3) = 0 yields x = ±8.
Once the critical points are found, we determine whether they are maxima or minima. This is done using the second derivative test. Taking the second derivative, we get f″(x) = 8x^(-5/3). We substitute the critical points into this second derivative. For x = 8, f″(8) = 1/4, which is positive. This indicates that the function is concave up at x = 8, confirming a relative minimum. For x = -8, f″(-8) is negative, so the function is concave down, confirming a relative maximum.
To complete the analysis, we plug the x-values back into the original function to find the corresponding y-values. f(8) = -48 and f(-8) = 48. These are the coordinates of the relative minimum and maximum, respectively.
Understanding relative extrema helps describe where the function increases or decreases most significantly. The relative maximum at (-8, 48) shows a local peak, while the relative minimum at (8, -48) shows a local dip in the graph. These points are important in applications like optimization and graph sketching.
