Question

given the function f(x)=x^3 - 9x^2 + 17x + 12, determine all coordinate points (x,y) on the graph of f such that the line tangent to f at (x,y) has a slope of 2. answer attempt 1 out of 2 two solutions and

Explanation:

Step1: Find the derivative of the function

The derivative of $f(x)=x^{3}-9x^{2}+17x + 12$ using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$ is $f^\prime(x)=3x^{2}-18x + 17$.

Step2: Set the derivative equal to the slope

Since the slope of the tangent line is 2, we set $f^\prime(x)=2$. So, $3x^{2}-18x + 17 = 2$.

Step3: Rearrange the equation to standard quadratic form

Subtract 2 from both sides to get $3x^{2}-18x+15 = 0$. Divide through by 3 to simplify: $x^{2}-6x + 5=0$.

Step4: Solve the quadratic equation

Factor the quadratic equation: $(x - 1)(x - 5)=0$. Using the zero - product property, $x-1 = 0$ or $x - 5=0$. So, $x = 1$ or $x = 5$.

Step5: Find the corresponding y - values

When $x = 1$, $y=f(1)=1^{3}-9\times1^{2}+17\times1 + 12=1-9 + 17+12=21$.
When $x = 5$, $y=f(5)=5^{3}-9\times5^{2}+17\times5 + 12=125-225+85 + 12=-3$.

Answer:

$(1,21)$ and $(5,-3)$