Consider the following cubic curve:
f(x)=−x3 −x2 +16x+16
1. Calculate f′(x).
2. Calculate f′′(x).
3. Find the x values such that f(x) = 0.
just use the [power rule to find the derivative of each term:
d/dx x^n = n*x^(n-1)
As for finding the roots, derivatives don't help much with that. But, digging back into your Algebra I skill set, note that
f(x) = -x^2(x+1) + 16(x+1)
= (16-x^2)(x+1)
...
To calculate the derivative of a function, such as f(x) = −x^3 − x^2 + 16x + 16, you can use the power rule. The power rule states that if you have a term of the form ax^n, the derivative is given by nax^(n-1).
1. Calculate f'(x):
Using the power rule, you differentiate each term of the function f(x) and simplify:
f'(x) = -3x^2 - 2x + 16.
2. Calculate f''(x):
To find the second derivative, you differentiate f'(x) using the power rule again:
f''(x) = -6x - 2.
3. Find x values such that f(x) = 0:
To find the x values where f(x) equals zero (i.e., the x-intercepts or roots of the cubic function), you set f(x) = 0 and solve for x. In this case, we have:
−x^3 − x^2 + 16x + 16 = 0.
To find the roots of this cubic equation, you can use various methods such as factoring, synthetic division, or numerical methods like Newton's method. In this case, since the equation is not easily factorable, we can use numerical methods or a graphing calculator to approximate the roots.
By analyzing the equation and using a graphing calculator, we can determine that the cubic curve intersects the x-axis at three distinct points, which correspond to the three x values where f(x) = 0.