Let f(x)=x^4-4x^3+10
Find the first derivative of the function. Find the critical values of the first derivative.
f'=4x^3-12x^2=4x^2(x-3)
setting that to zero, x=0, x=3
Thank you that the answer I got but wasn't quite sure if it was right.
To find the first derivative of the function f(x), you need to apply the power rule and the sum rule of derivatives. Here are the steps:
Step 1: Differentiate each term of the function f(x) separately:
- The derivative of x^4 with respect to x is 4x^(4-1) which simplifies to 4x^3.
- The derivative of -4x^3 with respect to x is -4*3x^(3-1) which simplifies to -12x^2.
- The derivative of 10 with respect to x is 0 since 10 is a constant.
Step 2: Combine the derivatives of each term to find the first derivative of f(x):
- The first derivative of f(x) is f'(x) = 4x^3 - 12x^2.
To find the critical values of the first derivative, we need to set f'(x) equal to zero and solve for x:
4x^3 - 12x^2 = 0
Step 3: Factor out the common term, 4x^2:
4x^2(x - 3) = 0
Step 4: Set each factor equal to zero and solve for x:
4x^2 = 0 => x = 0
x - 3 = 0 => x = 3
Therefore, the critical values of the first derivative are x = 0 and x = 3.