for what value(s) of x does the slope of the curve y=-x^3+3x^2+1 take on its largest value
slope=-3x^2+5x
dslope/dx=0=-6x+5
try x=5/6
Or :-) since slope = -3x^2 + 6x,
dslope/dx = 0 when -6x+6 = 0
or, x=1
To find the value(s) of x for which the slope of the curve y = -x^3 + 3x^2 + 1 is the largest, we need to determine where the derivative of the function is zero or undefined.
Step 1: Find the derivative of the function.
The given function is y = -x^3 + 3x^2 + 1. To find its derivative, we differentiate each term separately:
dy/dx = d/dx(-x^3) + d/dx(3x^2) + d/dx(1)
= -3x^2 + 6x
Step 2: Find the critical points.
Set the derivative equal to zero and solve for x:
-3x^2 + 6x = 0
Factor out x:
x(-3x + 6) = 0
Using the zero product property, we get:
x = 0 or -3x + 6 = 0
For the second equation, solve for x:
-3x + 6 = 0
-3x = -6
x = 2
So, the critical points are x = 0 and x = 2.
Step 3: Analyze the endpoints of the domain.
Since we have a polynomial function, there are no endpoints. Thus, we can skip this step.
Step 4: Determine which point(s) give the largest slope.
To determine whether x = 0, x = 2, or both give the largest slope, we can use the second derivative test. However, since we are looking for the largest value of the slope, we only need to compare the values of the slope at these points.
Evaluate the slope of the curve at each critical point:
Slope at x = 0: dy/dx = -3(0)^2 + 6(0) = 0
Slope at x = 2: dy/dx = -3(2)^2 + 6(2) = -12 + 12 = 0
The slopes at both points are zeros.
Therefore, there are no specific value(s) of x for which the slope of the curve y = -x^3 + 3x^2 + 1 takes on its largest value. The slope is always zero at every point on the curve.