Let f be the function defined by

Question

 f(x) = x^3-x^2-4x+4? 

Find the x- and y-intercepts of f. 
Write an equation of the line tangent to the graph of f at the y-intercept. 
Find the x-value(s) where the slope of the line tangent to the graph of f is zero. 

Answer 1

The function f(x) = x³ - x² - 4x + 4 

y = x³ - x² - 4x + 4

To find y intercept substitute x = 0 in y = x³ - x² - 4x + 4.

y = (0)³ - (0)² - 4(0) + 4

y = 4

y intercept of f(x) is (0, 4)

Real zeros are x intercepts of function.

f(x) = x³ - x² - 4x + 4

Identify Rational Zeros  

Usually it is not practical to test all possible zeros of a polynomial function using only synthetic substitution. The Rational Zero Theorem can be used for finding the some possible zeros to test.

f(x) = x³ - x² - 4x + 4

If p/q is a rational zero, then p is a factor of 4 and q is a factor of 1.

The possible values of p are   ± 1,   ± 2, and   ± 4.

The possible values for q are ± 1.

So, p/q = ± 1,   ± 2, ± 4.

Make a table for the synthetic division and test possible  zeros.

p/q	1	-1	-4	4
-1	1	-2	-2	6
1	1	0	-4	0

Since f(1) = 0,   x = 1 is a zero. The depressed polynomial is   x²- 4 = 0

 Find the roots of the related quadratic equation x²- 4 = 0

x² = 4

x = ± 2

Real zeros of f(x) = 2,-2 and 1

x intercepts f(x) are (2, 0), (-2, 0) and (1, 0).

Answer 2

The curve f(x) = x³ - x² - 4x + 4 

y' = 3x² - 2x - 4

y' is the slope of tangent line.

Equate y' = 0

3x² - 2x - 4 = 0

Compare it to ax² + bx + c = 0 and solve for x.

Roots are

The x values are 1.53 and - 0.86.

Answer 3

The curve y = x³ - x² - 4x + 4 

Differentiating on each side with respect of x .

y' = 3x² - 2x - 4

The tangent point is y intercept of curve (0, 4)

Substitute the values x = 0 in y'.

y' = 3(0)² - 2(0) - 4

y' = - 4

This is the slope of tangent line to the curve at (0, 4).

m = - 4

To find the tangent line equation, substitute the values of m = - 4 and (x, y ) = (0, 4).  in the slope intercept form of an equation y = mx + b.

4 = - 4(0) + b

4 = b

Substitute m = - 4 and b = 4 in y = mx + b.

y = - 4x + 4

Tangent line is y = - 4x + 4.