Question

Let $f(x)=(8\sin x+15\cos x+3)^{2}-15$, $x\in\mathbb{R}.$ Then the maximum value of $f$ is

• A. 325
• B. 365
• C. 385
• D. 430
• E. 455

Hint:

Logic Tip: When looking for the maximum of a squared term like $(u+c)^2$, you want $u$ to be the same sign as $c$ and as large as possible. Since $c$ is positive (+3), we choose the positive maximum bound for $u$ (+17).

Answer 1

The Correct Option is C

Concept:
The expression $a\sin x + b\cos x$ has a well-known maximum and minimum value over the real numbers. The range of $a\sin x + b\cos x$ is always: $$\left[-\sqrt{a^2+b^2}, \sqrt{a^2+b^2}\right]$$
Step 1: Find the range of the trigonometric component.
Let $u = 8\sin x + 15\cos x$. We identify $a = 8$ and $b = 15$. The maximum and minimum bounds for $u$ are: $$\pm\sqrt{8^2 + 15^2} = \pm\sqrt{64 + 225} = \pm\sqrt{289} = \pm17$$ So, $-17 \le u \le 17$.
Step 2: Substitute $u$ into the function $f(x)$.
Rewrite the original function in terms of $u$: $$f(x) = (u + 3)^2 - 15$$
Step 3: Maximize the quadratic function.
We need to find the maximum value of $(u + 3)^2 - 15$ subject to the domain $-17 \le u \le 17$. Since $(u + 3)^2$ is a parabola opening upwards, its maximum on a closed interval will occur at one of the endpoints. Let's test the bounds: If $u = -17$: $$(-17 + 3)^2 - 15 = (-14)^2 - 15 = 196 - 15 = 181$$ If $u = 17$: $$(17 + 3)^2 - 15 = (20)^2 - 15 = 400 - 15 = 385$$
Step 4: Identify the maximum value.
Comparing the outcomes, the maximum value is $385$.