Find Increasing Intervals for the Quadratic Function y = (3+x)(x-7)

To solve this problem, follow these steps:

• Step 1: **Expand the Function**
  The given function is $y = (3+x)(x-7)$. Expanding it gives: ${y = x^2 - 4x - 21.}$
• Step 2: **Find the Derivative**
  Compute the derivative of $y$: $y' = 2x - 4.$
• Step 3: **Find Critical Points**
  Set the derivative equal to zero and solve: $2x - 4 = 0 \Rightarrow x = 2.$ This is a critical point where the slope changes from negative to positive or vice versa.
• Step 4: **Test Intervals**
  Choose test points around the critical point $x = 2$ to determine the sign of $y'$:
  For $x < 2$, e.g., $x = 0$: $y'(0) = 2(0) - 4 = -4$ (negative).
  For $x > 2$, e.g., $x = 3$: $y'(3) = 2(3) - 4 = 2$ (positive).

Hence, the function is increasing for $x > 2$.

Thus, the function increases in the interval $x > 2$.