Question

For what value of the constant c is the function f continuous on (-[infinity], [infinity])? f(x) = { cx²/8x if x < 5, x³ - cx if x ≥ 5

Answer 1

Finall Answer:

The function
`\( f \)` is continuous on
`\((- \infty, \infty) \) when \( c = 20 \).`

Step-by-step explanation:

For the function
`\( f(x) = \begin{cases} (cx^2)/(8x) & \text{if } x < 5 \\ x^3 - cx & \text{if } x \geq 5 \end{cases} \) to be continuous at \( x = 5 \), the`limits from both sides of the function must be equal.

For
`\( x < 5 \), \( \lim_{{x \to 5^-}} (cx^2)/(8x) = (25c)/(8) \). For \( x \geq 5 \), \( \lim_{{x \to 5^+}} x^3 - cx = 5^3 - 5c = 125 - 5c \). To ensure`continuity at
`\( x = 5 \),` these two limits must be equal, so
`\( (25c)/(8) = 125` - 5c \). Solving this equation yields
`\( c = 20 \),`which ensures the function
`\( f \) is continuous on \((- \infty, \infty) \).`