Analyzing the Critical Points and Determining Relative Maximum: How to Find the Value of x for which the Function g Attains a Relative Maximum in Math

Let g be the function defined by g(x)=∫x−1(−12+cos(t3+2t))ⅆt for 0

To find the value of x at which the function g attains a relative maximum, we need to find the critical points of the function and then determine whether they are relative maximums or minimums

To find the value of x at which the function g attains a relative maximum, we need to find the critical points of the function and then determine whether they are relative maximums or minimums.

First, let’s find the derivative of g(x). Since g(x) is defined as an integral, we can apply the Fundamental Theorem of Calculus to find the derivative:

g'(x) = (-12 + cos(x^3 + 2x))/(x^(-1))

Next, let’s find the critical points by setting g'(x) equal to zero:

(-12 + cos(x^3 + 2x))/(x^(-1)) = 0

To simplify this expression, we can multiply both sides by x:

-12 + cos(x^3 + 2x) = 0

Next, add 12 to both sides:

cos(x^3 + 2x) = 12

Since the range of cosine function is -1 to 1, there are no real values of x that satisfy this equation. Therefore, there are no critical points for g'(x) and, consequently, no relative maximums for g(x).

In other words, the function g(x) does not have a relative maximum within the given interval (0, π/2).

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