Let $\sigma_{\mathrm{imp}}(K)$ denote the implied volatility of a call option with strike price $K$, defined from the relation

$$
M_{C}(K, S, r, \tau)=C\left(K, S, \sigma_{\mathrm{imp}}(K), r, \tauight) \text {, }
$$

where $M_{C}$ is the market price of the call option, $C\left(K, S, \sigma_{\mathrm{imp}}(K), r, \tauight)$ is the Black-Scholes call pricing function, $S$ is the underlying asset price, $\tau$ is the time remaining until maturity, and $r$ is the risk-free interest rate.

a) Compute the partial derivative

$$
\frac{\partial M_{C}}{\partial K}(K, S, r, \tau)
$$

using the functions $C$ and $\sigma_{\text {imp }}$.

b) Knowing that market call option prices $M_{C}(K, S, r, \tau)$ are decreasing in the strike prices $K$, find an upper bound for the slope $\sigma_{\mathrm{imp}}^{\prime}(K)$ of the implied volatility curve.

c) Similarly, knowing that the market put option prices $M_{P}(K, S, r, \tau)$ are increasing in the strike prices $K$, find a lower bound for the slope $\sigma_{\mathrm{imp}}^{\prime}(K)$ of the implied volatility curve.