Question

find the value of a that makes the following function differentiable for all values of x.
g(x) = {ax if x < 0; x^2 - 9x if x >= 0}
the function g(x) is differentiable for all values of x when a = . (simplify your answer.)

Explanation:

Step1: Check left - hand and right - hand derivatives at x = 0

For \(x<0\), \(g(x)=ax\), so \(g'(x)=a\). For \(x\geq0\), \(g(x)=x^{2}-9x\), so \(g'(x)=2x - 9\).

Step2: Evaluate right - hand derivative at x = 0

Substitute \(x = 0\) into \(g'(x)=2x - 9\). We get \(g'(0)=2\times0 - 9=-9\).

Step3: Set left - hand and right - hand derivatives equal

Since the function is differentiable at \(x = 0\), the left - hand derivative and the right - hand derivative at \(x = 0\) must be equal. So \(a=-9\).

Answer:

\(-9\)