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William, a logistician, needs to route a freight train that is 20 feet at its tallest point and 10 feet at its widest point within 3 days. The most direct path includes a single-track tunnel that needs 24 hour notice prior to use. If the tunnel is roughly modeled by f(x)=-0.x^2+3.2x-3.5, should William make arrangements for the train to use the tunnel?

Verified Answer :

The minimum value of f(x) is at the solution of  and if the solution is x = a,  .

For f(x) = -1x^2 + 3.2x - 3.5, f'(x) = -0.2x + 3.2 and f''(x) = -0.2 which is negative for all values of x.

Solving f'(x) = 0 gives -0.2x + 3.2 = 0

=> x = 3.2/0.2 = 16

f(16) = 22.1

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