MHB Does anyone know the answer to this?

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The discussion revolves around a mathematical model for the power required to maintain an electric car's motion against drag, represented by the equation P = Av^2 + (B/v). Participants are tasked with finding the speed vP that minimizes power, calculating the corresponding power, and determining how far the car can travel with a given energy store E. The solution involves deriving the first derivative of power with respect to velocity, setting it to zero, and substituting back into the power equation. Additionally, the relationship between power, energy, and time is highlighted, emphasizing that time is maximized for a fixed energy at the optimal speed. The conversation suggests a need for clarity on the appropriate forum for such mathematical discussions.
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A simplified model of the power P required to sustain the motion of an electric car at speed v experiencing nonzero drag can be modeled by the equation:

P = Av^2 + (B/v)
(where A and B are positive constants.)

(a) What speed vP minimizes power?
(b) What power does the speed in (a) require?
(c) Suppose that an electric car has a usable store E of energy. How far dP can the electric car travel at the
speed found in (b)?
 
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Jimmy Perdon said:
A simplified model of the power P required to sustain the motion of an electric car at speed v experiencing nonzero drag can be modeled by the equation:

P = Av^2 + (B/v)
(where A and B are positive constants.)

(a) What speed vP minimizes power?
(b) What power does the speed in (a) require?
(c) Suppose that an electric car has a usable store E of energy. How far dP can the electric car travel at the
speed found in (b)?

(a) find $\dfrac{dP}{dv}$, set it equal to zero, and determine the value of velocity that minimizes power.

(b) substitute the value of velocity found in part (a) into the original power equation

(c) Power is a rate of energy use over time. At the velocity found in part (a), time will be a maximum for a fixed value of available energy, E.

$P = \dfrac{E}{t} \implies t = \dfrac{E}{P}$, and distance traveled is $d = v \cdot t$why is this posted in the chat room? should be in calculus.
 
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Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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