Questions: Find values of a and b that make the function provided below both continuous and differentiable at 3. f(x) = ax+b for x<3 x^2 for x ≥ 3 a = Number b = Number

Find values of a and b that make the function provided below both continuous and differentiable at 3.
f(x) =
ax+b for x<3
x^2 for x ≥ 3

a = Number
b = Number

Transcript text: Find values of $a$ and $b$ that make the function provided below both continuous and differentiable at 3 . \[ f(x)=\left\{\begin{array}{ll} a x+b & x<3 \\ x^{2} & x \geq 3 \end{array}\right. \] a=\text { Number } \\ b=\text { Number }

Solution

Solution Steps

Step 1: Define the Function Pieces

We define the piecewise function $ f(x) $ as follows: \[ f(x) = \begin{cases} a x + b & \text{if } x < 3 \\ x^2 & \text{if } x \geq 3 \end{cases} \]

Step 2: Establish the Continuity Condition

To ensure continuity at $ x = 3 $, we set the left-hand limit equal to the right-hand limit: \[ \lim_{{x \to 3^-}} f(x) = \lim_{{x \to 3^+}} f(x) = f(3) \] This gives us the equation: \[ 3a + b = 9 \]

Step 3: Establish the Differentiability Condition

To ensure differentiability at $ x = 3 $, we set the derivative from the left-hand side equal to the derivative from the right-hand side: \[ \lim_{{x \to 3^-}} f'(x) = \lim_{{x \to 3^+}} f'(x) \] This results in the equation: \[ a = 6 \]

Step 4: Solve the System of Equations

We now have a system of equations: