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(a). Consider the parabolic function f(x) = ax² +bx+c, where a ±0, b and care constants. For what values of a, band c is f

(i) concave up?

(ii) concave down?

[Verify your answer by MATHEMATICA and attach the printout of the commands and output]

A Consider The Parabolic Function Fx Ax Bxc Where A 0 B And Care Constants For What Values Of A Band C Is F I Concave Up Ii Concave Down Verify Your Answer By M class=

Sagot :

LammettHash

Information about concavity is contained in the second derivative of a function. Given f(x) = ax² + bx + c, we have

f'(x) = 2ax + b

and

f''(x) = 2a

Concavity changes at a function's inflection points, which can occur wherever the second derivative is zero or undefined. In this case, since a ≠ 0, the function's concavity is uniform over its entire domain.

(i) f is concave up when f'' > 0, which occurs when a > 0.

(ii) f is concave down when f'' < 0, and this is the case if a < 0.

In Mathematica, define f by entering

f[x_] := a*x^2 + b*x + c

Then solve for intervals over which the second derivative is positive or negative, respectively, using

Reduce[f''[x] > 0, x]

Reduce[f''[x] < 0, x]

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