Question

If \(f(x)=x^2−7x\) and \(g(x)=x+3\), then the minimum value of \(f(g(x))−3x\) is

• A. -20
• B. -15
• C. -12
• D. -16

Answer 1

The Correct Option is D

We are given the following functions: 

• \(f(x) = x^2 - 7x\)
• \(g(x) = x + 3\)

First, we need to find the expression for \( f(g(x)) \). Substituting \( g(x) \) into \( f(x) \):

\(f(g(x)) = (x + 3)^2 - 7(x + 3)\)

Expanding the terms:

\(f(g(x)) = x^2 + 6x + 9 - 7x - 21\)

Now, simplify the expression:

\(f(g(x)) = x^2 - x - 12\)

Now, we are interested in \( f(g(x)) - 3x \):

\(f(g(x)) - 3x = x^2 - x - 12 - 3x\)

Simplifying this further:

\(f(g(x)) - 3x = x^2 - 4x - 12\)

To find the minimum value of this expression, we can find the derivative and set it to zero.

First, take the derivative of \( f(g(x)) - 3x \):

\(\frac{d}{dx} (x^2 - 4x - 12) = 2x - 4\)

Now, set the derivative equal to zero to find the critical point:

\(2x - 4 = 0\)

Solve for \( x \):

\(x = 2\)

Now, substitute \( x = 2 \) into the expression \( f(g(x)) - 3x \) to find the minimum value:

\(f(g(2)) - 3(2) = 2^2 - 4(2) - 12 = 4 - 8 - 12 = -16\)

Thus, the minimum value of \( f(g(x)) - 3x \) is -16.

Answer 2

We have:

\(f(g(x)) - 3x\) 

\(\Rightarrow f(x+3) - 3x\)

\(= (x+3)^2 - 7(x+3) - 3x\)

\(= x^2 - 4x - 12\)

Now, let’s find the value of the expression:

\(-\frac{D}{4a} \) where \( \frac{(4ac - b^2)}{4a}\)

Substituting the values:

\(= - \frac{(16 + 48)}{4}\)

\(= -16\)