1. **Stating the problem:** We are given the equation $x^5 = 2x^3 + c$ and a point $(3,5)$ on the curve. We need to find the constant $c$ and verify the gradient (derivative) at $x=3$ is 2. 2. **Find $c$ using the point:** Substitute $x=3$ and $y=5$ into the equation: $$5 = 3^5 - 2 \times 3^3 + c$$ Calculate powers: $$5 = 243 - 54 + c$$ Simplify: $$5 = 189 + c$$ Solve for $c$: $$c = 5 - 189 = -184$$ 3. **Find the gradient (derivative) of $y = x^5 - 2x^3 + c$:** $$\frac{dy}{dx} = 5x^4 - 6x^2$$ 4. **Evaluate the gradient at $x=3$:** $$\frac{dy}{dx}\bigg|_{x=3} = 5 \times 3^4 - 6 \times 3^2 = 5 \times 81 - 6 \times 9 = 405 - 54 = 351$$ 5. **Check gradient condition:** The problem states the gradient is 2 at $x=3$, but our calculation shows 351. This suggests either the gradient condition or the original equation might be misstated. **Final answer:** - The constant $c$ is $-184$. - The gradient at $x=3$ is $351$, not $2$. Please verify the problem statement if the gradient must be 2 at $x=3$.