QuestionFind the value of $d$ if the minimum of $f(x) = \cos x + d$ is -6.

Studdy Solution

STEP 1

Assumptions1. The function is $f(x)=\cos x+d$ . The minimum value of the function is -6

STEP 2

We know that the minimum value of the cosine function $\cos x$ is -1. This occurs at $x=\pi,\pi,5\pi, \ldots$ and so on.
$\min(\cos x) = -1$

STEP 3

Since $f(x)=\cos x+d$ , the minimum value of $f(x)$ is achieved when $\cos x$ is at its minimum. Therefore, the minimum value of $f(x)$ is $-1+d$ .
$\min(f(x)) = \min(\cos x) + d = -1 + d$

STEP 4

We are given that the minimum value of $f(x)$ is -6. Therefore, we can set up the equation $-1+d=-6$ .
$-1 + d = -6$

STEP 5

olve the equation for $d$ .
$d = - +1$

STEP 6

Calculate the value of $d$ .
$d = -6 +1 = -5$ The value of $d$ is -5.

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