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Find Relative Maxima and Minima of Cubic Function f(x)=2x³−3x²−36x+4
Mathematics
Grade 11 of Senior High School
Question Content
The graph and equation of the function f are given. a. Use the graph to find any values at which f has a relative maximum, and use the equation to calculate the relative maximum for each value. b. Use the graph to find any values at which f has a relative minimum, and use the equation to calculate the relative minimum for each value. The function is f(x)=2x³−3x²−36x+4, with the viewing window [−5,5,1] by [−100,100,10]. For part a, select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The function f has (a) relative maxima(maximum) at ____ and the relative maxima(maximum) are(is) ____. (Use a comma to separate answers as needed.) B. The function f has no relative maxima.
Correct Answer
a. Option A: The function f has a relative maximum at -2 and the relative maximum is 48; b. The function f has a relative minimum at 3 and the relative minimum is -77
Detailed Solution Steps
1
Step 1: Find the critical points using calculus (to confirm the graph's values). Take the first derivative of f(x): f'(x) = 6x² - 6x - 36. Set f'(x)=0: 6x² -6x -36=0, divide by 6: x² -x -6=0. Factor the quadratic: (x+2)(x-3)=0, so critical points are x=-2 and x=3.
2
Step 2: Verify the relative maximum from the graph and calculate its value. The graph shows a peak at x=-2. Substitute x=-2 into f(x): f(-2)=2(-2)³−3(-2)²−36(-2)+4 = 2(-8) -3(4) +72 +4 = -16-12+72+4=48. So this is a relative maximum.
3
Step 3: Verify the relative minimum from the graph and calculate its value. The graph shows a valley at x=3. Substitute x=3 into f(x): f(3)=2(3)³−3(3)²−36(3)+4 = 2(27)-3(9)-108+4=54-27-108+4=-77. So this is a relative minimum.
4
Step 4: Select the correct option for part a: Option A, with x=-2 and value 48.
Knowledge Points Involved
1
Relative Extrema of Polynomial Functions
Relative maxima (peaks) and minima (valleys) are points on a function where the function changes from increasing to decreasing (maxima) or decreasing to increasing (minima). They can be identified visually from a graph or calculated using calculus.
2
First Derivative Test for Critical Points
For a differentiable function f(x), critical points occur where f'(x)=0 or f'(x) is undefined. For polynomial functions, we set the first derivative equal to zero, solve for x to find critical points, then test intervals around them to classify as maxima, minima, or neither.
3
Evaluating Polynomial Functions
To find the value of a polynomial at a given x-value, substitute the x-value into the polynomial expression and simplify using order of operations (PEMDAS/BODMAS).
4
Reading Function Graphs
When given a graph of a function, relative maxima are the highest points in their local interval, and relative minima are the lowest points in their local interval. The x-coordinate is the input value, and the y-coordinate is the function's output value at that point.
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