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$ifferentiability - System of Equations Score: 4//5 Penalty: none Question Show Examples Determine the values of a and b which would result in the function f(x) being differentiable at x=-1. f(x)={[ax-3," for ",x <= -1],[2bx^(2)-4x+9," for ",x > -1]:} Answer Attempt 1 out of 2 a=◻quad b=◻quad" Submit Answer "$

ifferentiability - System of Equations $\newline$ Score: $4 / 5$ $\newline$ Penalty: none $\newline$ Question $\newline$ Show Examples $\newline$ Determine the values of $a$ and $b$ which would result in the function $f(x)$ being differentiable at $x=-1$ . $\newline$ $f(x)=\left\{\begin{array}{lll} a x-3 & \text { for } & x \leq-1 \\ 2 b x^{2}-4 x+9 & \text { for } & x>-1 \end{array}\right.$ $\newline$ Answer Attempt $1$ out of $2$ $\newline$ $a=\square \quad b=\square \quad \text { Submit Answer }$

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Q. ifferentiability - System of Equations

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Score:

4 / 5

\newline

Penalty: none

\newline

Question

\newline

Show Examples

\newline

Determine the values of

a

and

b

which would result in the function

f(x)

being differentiable at

x=-1

\newline

f(x)=\left\{\begin{array}{lll} a x-3 & \text { for } & x \leq-1 \\ 2 b x^{2}-4 x+9 & \text { for } & x>-1 \end{array}\right.

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Answer Attempt

1

out of

2

\newline

a=\square \quad b=\square \quad \text { Submit Answer }

Derivatives for $x \leq -1$ : For $f(x)$ to be differentiable at $x=-1$ , the left-hand limit and right-hand limit of $f'(x)$ must be equal at $x=-1$ .
Derivatives for $x > -1$ : First, find the derivative of $ax - 3$ for $x \leq -1$ , which is $f'(x) = a$ .
Derivative Equality at $x = -1$ : Next, find the derivative of $2bx^2 - 4x + 9$ for $x > -1$ , which is $f'(x) = 4bx - 4$ .
Solving for $a$ : Set the derivatives equal to each other at $x = -1$ : $a = 4b(-1) - 4$ .
Ensuring Continuity at $x = -1$ : Solve for $a$ : $a = -4b - 4$ .
Substitute $x = -1$ : Now, ensure the function itself is continuous at $x = -1$ by setting the left-hand side equal to the right-hand side: $ax - 3 = 2bx^2 - 4x + 9$ at $x = -1$ .
Solving for b: Plug in $x = -1$ : $a(-1) - 3 = 2b(-1)^2 - 4(-1) + 9$ .
Calculating a: Simplify the equation: $-a - 3 = 2b + 4 + 9$ .
Final Simplification: Further simplify: $-a - 3 = 2b + 13$ .
Final Simplification: Further simplify: $-a - 3 = 2b + 13$ .Now, substitute the value of $a$ from the derivative equality: $-(-4b - 4) - 3 = 2b + 13$ .
Final Simplification: Further simplify: $-a - 3 = 2b + 13$ .Now, substitute the value of $a$ from the derivative equality: $-(-4b - 4) - 3 = 2b + 13$ .Simplify: $4b + 4 - 3 = 2b + 13$ .
Final Simplification: Further simplify: $-a - 3 = 2b + 13$ .Now, substitute the value of $a$ from the derivative equality: $-(-4b - 4) - 3 = 2b + 13$ .Simplify: $4b + 4 - 3 = 2b + 13$ .Combine like terms: $4b + 1 = 2b + 13$ .
Final Simplification: Further simplify: $-a - 3 = 2b + 13$ .Now, substitute the value of $a$ from the derivative equality: $-(-4b - 4) - 3 = 2b + 13$ .Simplify: $4b + 4 - 3 = 2b + 13$ .Combine like terms: $4b + 1 = 2b + 13$ .Subtract $2b$ from both sides: $2b + 1 = 13$ .
Final Simplification: Further simplify: $-a - 3 = 2b + 13$ .Now, substitute the value of $a$ from the derivative equality: $-(-4b - 4) - 3 = 2b + 13$ .Simplify: $4b + 4 - 3 = 2b + 13$ .Combine like terms: $4b + 1 = 2b + 13$ .Subtract $2b$ from both sides: $2b + 1 = 13$ .Subtract $1$ from both sides: $2b = 12$ .
Final Simplification: Further simplify: $-a - 3 = 2b + 13$ .Now, substitute the value of $a$ from the derivative equality: $-(-4b - 4) - 3 = 2b + 13$ .Simplify: $4b + 4 - 3 = 2b + 13$ .Combine like terms: $4b + 1 = 2b + 13$ .Subtract $2b$ from both sides: $2b + 1 = 13$ .Subtract $1$ from both sides: $2b = 12$ .Divide by $2$ to solve for $a$ $0$ : $a$ $1$ .
Final Simplification: Further simplify: $-a - 3 = 2b + 13$ .Now, substitute the value of $a$ from the derivative equality: $-(-4b - 4) - 3 = 2b + 13$ .Simplify: $4b + 4 - 3 = 2b + 13$ .Combine like terms: $4b + 1 = 2b + 13$ .Subtract $2b$ from both sides: $2b + 1 = 13$ .Subtract $1$ from both sides: $2b = 12$ .Divide by $2$ to solve for $a$ $0$ : $a$ $1$ .Now, plug $a$ $0$ back into the equation for $a$ : $a$ $4$ .
Final Simplification: Further simplify: $-a - 3 = 2b + 13$ .Now, substitute the value of $a$ from the derivative equality: $-(-4b - 4) - 3 = 2b + 13$ .Simplify: $4b + 4 - 3 = 2b + 13$ .Combine like terms: $4b + 1 = 2b + 13$ .Subtract $2b$ from both sides: $2b + 1 = 13$ .Subtract $1$ from both sides: $2b = 12$ .Divide by $2$ to solve for $a$ $0$ : $a$ $1$ .Now, plug $a$ $0$ back into the equation for $a$ : $a$ $4$ .Calculate $a$ : $a$ $6$ .
Final Simplification: Further simplify: $-a - 3 = 2b + 13$ .Now, substitute the value of $a$ from the derivative equality: $-(-4b - 4) - 3 = 2b + 13$ .Simplify: $4b + 4 - 3 = 2b + 13$ .Combine like terms: $4b + 1 = 2b + 13$ .Subtract $2b$ from both sides: $2b + 1 = 13$ .Subtract $1$ from both sides: $2b = 12$ .Divide by $2$ to solve for $a$ $0$ : $a$ $1$ .Now, plug $a$ $0$ back into the equation for $a$ : $a$ $4$ .Calculate $a$ : $a$ $6$ .Simplify to find $a$ : $a$ $8$ .