Let f(x)=x^4-4x^3+10

Find the first derivative of the function. Find the critical values of the first derivative.

To find the first derivative of the function f(x) = x^4 - 4x^3 + 10, we need to use the power rule for differentiation.

The power rule states that if f(x) = x^n, where n is a constant, then the derivative of f(x) is given by f'(x) = nx^(n-1).

Applying the power rule to each term of the function, we have:

f'(x) = d/dx (x^4) - d/dx (4x^3) + d/dx (10)

For the first term, the power rule gives us 4x^(4-1) = 4x^3.
For the second term, the power rule gives us 3(4x^(3-1)) = 12x^2.
For the third term, the derivative of a constant is zero, so it disappears.

Combining the derivatives of each term, we have:

f'(x) = 4x^3 - 12x^2

Now, we need to find the critical values of the first derivative.

Critical values occur when the derivative is either zero or undefined. To find the critical values, we set f'(x) equal to zero and solve for x:

4x^3 - 12x^2 = 0

Factoring out the common term, we have:

4x^2(x - 3) = 0

Setting each factor equal to zero, we get:

4x^2 = 0 --> x^2 = 0 --> x = 0

x - 3 = 0 --> x = 3

So, the critical values of the first derivative are x = 0 and x = 3.