1. **Problem:** Given the function $f(q) = hq^2 + mq + c$ with gradient function $4q + 8$ and a stationary value of $-3$, find $h$, $m$, and $c$. 2. **Step 1: Understand the gradient function** The gradient (derivative) of $f(q)$ is: $$f'(q) = \frac{d}{dq}(hq^2 + mq + c) = 2hq + m$$ 3. **Step 2: Match the given gradient function** We know $f'(q) = 4q + 8$, so: $$2hq + m = 4q + 8$$ Equate coefficients: - Coefficient of $q$: $2h = 4 \Rightarrow h = 2$ - Constant term: $m = 8$ 4. **Step 3: Use the stationary value condition** A stationary point occurs where $f'(q) = 0$. Set $4q + 8 = 0$: $$4q + 8 = 0 \Rightarrow q = -2$$ 5. **Step 4: Use the stationary value $f(-2) = -3$** Substitute $q = -2$ into $f(q)$: $$f(-2) = h(-2)^2 + m(-2) + c = 2 \times 4 + 8 \times (-2) + c = 8 - 16 + c = -8 + c$$ Set equal to $-3$: $$-8 + c = -3 \Rightarrow c = 5$$ 6. **Final answer:** $$h = 2, \quad m = 8, \quad c = 5$$