Question

A quadratic function is given. f(x) = 1 + x − 7 x2 (a) Use a graphing device to find the maximum or minimum value of the quadratic function f, rounded to two decimal places. The Correct: Your answer is correct. value of f is f(x) = 1.09 Correct: Your answer is correct. . (b) Find the exact maximum or minimum value of f, and compare it with your answer to part (a).

Answer 1

Answer:

a) $P (x,y) \approx (0.1,1.05)$, b) $P(x,y) = (0.07,1.04)$.

Step-by-step explanation:

a) The graphic is enclosed to the problem. By visual inspection, an absolute maximum is found.

$P (x,y) \approx (0.1,1.05)$

b) The exact method consists in the application of the First and Second Derivative Tests. First and second derivatives are, respectively:

$f'(x) = 1 - 14\cdot x$

$f''(x) = -14$

The First Derivative Test consists in equalizing the first derivative to zero and solving the expression:

$1 - 14\cdot x = 0$

$x = 0.07$

According to the second derivative, the critical point leads to a maximum. The remaining component is determined by evaluation the polynomial:

$y = 1 +0.07-7\cdot (0.07)^{2}$

$y = 1.04$

The exact solution is $P(x,y) = (0.07,1.04)$, indicating that graphic solution leads to a good approximation.