what value of the constant c would make the function f continuous on the interval (-∞,∞)? f(x) = {x² - 5, x…

what value of the constant c would make the function f continuous on the interval (-∞,∞)? f(x) = {x² - 5, x ≤ c; 8x - 21, x > c}

what value of the constant c would make the function f continuous on the interval (-∞,∞)? f(x) = {x² - 5, x ≤ c; 8x - 21, x > c}

Answer

Explanation:

Step1: Recall continuity condition

For a function to be continuous at $x = c$, $\lim_{x\rightarrow c^{-}}f(x)=\lim_{x\rightarrow c^{+}}f(x)=f(c)$. So we set $x^{2}-5 = 8x - 21$ when $x = c$.

Step2: Rearrange the equation

Set up the quadratic - equation: $c^{2}-8c + 16=0$. This is obtained by moving all terms to one side of the equation $c^{2}-5=8c - 21$, so $c^{2}-8c+16 = 0$.

Step3: Factor the quadratic equation

The quadratic equation $c^{2}-8c + 16=(c - 4)^{2}=0$.

Step4: Solve for $c$

Taking the square - root of both sides of $(c - 4)^{2}=0$, we get $c = 4$.

Answer:

$4$