the function f is given by f(x)=x^3 - 8x^2 + 20x - 12. for how many positive values of c does…
the function f is given by f(x)=x^3 - 8x^2 + 20x - 12. for how many positive values of c does lim(x→c)f(x)=4? (a) 0 (b) 1 (c) 2 (d) 3 (e) 4
Answer
Explanation:
Step1: Set up the equation
Since $\lim_{x\rightarrow c}f(x)=f(c)$ for a polynomial function $f(x)=x^{3}-8x^{2}+20x - 12$, we set $f(c)=4$. So, $c^{3}-8c^{2}+20c - 12 = 4$.
Step2: Rearrange the equation
Rearrange to get $c^{3}-8c^{2}+20c-16 = 0$.
Step3: Try some simple positive - integer values
Let's try $c = 2$: $2^{3}-8\times2^{2}+20\times2 - 16=8 - 32+40 - 16=0$. So, $(c - 2)$ is a factor of $c^{3}-8c^{2}+20c - 16$.
Step4: Perform polynomial long - division
Dividing $c^{3}-8c^{2}+20c - 16$ by $(c - 2)$ gives $c^{2}-6c + 8$.
Step5: Factor the quadratic
Factor $c^{2}-6c + 8=(c - 2)(c - 4)$.
Step6: Find the roots
Set $(c - 2)(c - 2)(c - 4)=0$. The roots are $c = 2$ and $c = 4$.
Answer:
C. 2