Determine the nature of all critical points of f(x)=-3x^4+8x^3, then sketch a rough graph of the function. Thank you for the help!!
Critical points occur where the derivative of the function is equal to zero. In this case, the first derivative of f(x)=-3x^4+8x^3 is -12x^3+24x^2 which can be factored into (-12)(x^2)(x-2). This implies that our critical points when f(x)=0 are located at x=0, and x=2. If we plug in x=0 into the function, we see that our first critical point is (0,0) and if we plug in x=2 into the function, then our second critical point is (2,16). Therefore, your critical points for the function f(x)=-3x^4+8x^3 are (0,0) and (2,16).
slope = -12 x^3 + 24x^2
curvature = -36 x^2 + 48 x
so we need to sketch a graph
Where is y = 0?
y = x^3 (-3x +8)
when x = 0 and when x = 8/3
what happens when |x| is big ?
y = x^3 (-3x +8) ===> -3 x^4
well x^4 is a big positive number way left or way right of the origin
so y dives for - infinity way left and way right
Now where is the slope = 0 ?
slope = -12 x^3 + 24x^2 = x*2 (-12 x + 24) = 12 x^2 (-x+2)
That is zero at zero and at x = 2
so what is y when x = 2?
y = x^3 (-3x +8) ===> 8 (-6+8) = 16
so the function is horizontal at the origin and at (2,16)
is it a maximum at (2,16) ?
at x = 2 curvature = -36 x^2 + 48 x = -36*4+48*2 = -48
so that curves down at (2,16)
at the origin the curvature is zero, so it just pauses
now start sketching
To determine the nature of the critical points of the function f(x)=-3x^4+8x^3, we need to find the derivative of the function, set it to zero, and solve for x. The values of x that make the derivative zero or undefined will be the critical points.
Step 1: Find the derivative of f(x).
Taking the derivative of f(x) = -3x^4+8x^3, we apply the power rule for differentiation, which states that if we have a term of the form ax^n, the derivative will be nax^(n-1).
f'(x) = -12x^3 + 24x^2
Step 2: Set the derivative equal to zero and solve for x.
-12x^3 + 24x^2 = 0
Factor out a common term of 12x^2:
12x^2(-x + 2) = 0
This equation will be zero when either 12x^2 = 0 (giving x = 0) or -x + 2 = 0 (giving x = 2).
Therefore, the critical points are x = 0 and x = 2.
Step 3: Determine the nature of the critical points.
To determine the nature of each critical point, we can use the second derivative test. The second derivative, f''(x), will tell us if the critical point is a local maximum, minimum, or an inflection point.
To find the second derivative, we differentiate the first derivative, f'(x), using the power rule:
f''(x) = -36x^2 + 48x
Now, substitute the critical points x = 0 and x = 2 into the second derivative.
At x = 0:
f''(0) = -36(0)^2 + 48(0) = 0
At x = 2:
f''(2) = -36(2)^2 + 48(2) = -96
The nature of the critical points is determined as follows:
- When the second derivative is positive, the function has a local minimum at the critical point.
- When the second derivative is negative, the function has a local maximum at the critical point.
- When the second derivative is zero or undefined, the test is inconclusive, and the nature of the critical point remains uncertain.
In our case:
- At x = 0, f''(0) = 0, so the nature of the critical point at x = 0 is uncertain.
- At x = 2, f''(2) = -96, so the nature of the critical point at x = 2 is a local maximum.
Step 4: Sketch a rough graph of the function.
Using the information gathered, we can now sketch a rough graph of the function. We know that at x = 0, there is a critical point of uncertain nature, and at x = 2, there is a local maximum.
Based on the degree of the polynomial and the signs of its coefficients, we can observe that as x approaches positive or negative infinity, the function approaches negative infinity. Additionally, the leading coefficient (-3) determines that the graph opens downward.
Hence, the graph will have the following characteristics:
- The function decreases as x approaches negative infinity.
- The function has a local maximum at x = 2.
- The function increases as x approaches positive infinity.
Please note that a rough graph may not accurately represent the function's precise shape. It is always preferable to use graphing software or a graphing calculator for a more accurate plot of the function.