Tangent Line A25379

1. **State the problem:** We are given a line $y = 4x - 6\pi$ that is tangent to the graph of a function $f$ at $x = a$. We need to find which function $f$ and value $a$ from the options satisfy this condition. 2. **Recall the tangent line conditions:** - The slope of the tangent line at $x = a$ equals the derivative of $f$ at $a$: $$f'(a) = 4$$ - The tangent line passes through the point $(a, f(a))$, so: $$f(a) = 4a - 6\pi$$ 3. **Check each option:** **Option A:** $f(x) = -4 \sin x$, $a = \frac{\pi}{2}$ - Derivative: $$f'(x) = -4 \cos x$$ - Evaluate slope at $a$: $$f'\left(\frac{\pi}{2}\right) = -4 \cos \frac{\pi}{2} = -4 \times 0 = 0 \neq 4$$ - So slope condition fails. **Option B:** $f(x) = -4 \cos x$, $a = \frac{\pi}{2}$ - Derivative: $$f'(x) = 4 \sin x$$ - Evaluate slope at $a$: $$f'\left(\frac{\pi}{2}\right) = 4 \sin \frac{\pi}{2} = 4 \times 1 = 4$$ - Slope condition satisfied. - Check function value: $$f\left(\frac{\pi}{2}\right) = -4 \cos \frac{\pi}{2} = -4 \times 0 = 0$$ - Check tangent line value at $a$: $$4 \times \frac{\pi}{2} - 6\pi = 2\pi - 6\pi = -4\pi$$ - Since $f(a) = 0 \neq -4\pi$, point condition fails. **Option C:** $f(x) = 4 \sin x$, $a = \frac{3\pi}{2}$ - Derivative: $$f'(x) = 4 \cos x$$ - Evaluate slope at $a$: $$f'\left(\frac{3\pi}{2}\right) = 4 \cos \frac{3\pi}{2} = 4 \times 0 = 0 \neq 4$$ - Slope condition fails. **Option D:** $f(x) = 4 \cos x$, $a = \frac{3\pi}{2}$ - Derivative: $$f'(x) = -4 \sin x$$ - Evaluate slope at $a$: $$f'\left(\frac{3\pi}{2}\right) = -4 \sin \frac{3\pi}{2} = -4 \times (-1) = 4$$ - Slope condition satisfied. - Check function value: $$f\left(\frac{3\pi}{2}\right) = 4 \cos \frac{3\pi}{2} = 4 \times 0 = 0$$ - Check tangent line value at $a$: $$4 \times \frac{3\pi}{2} - 6\pi = 6\pi - 6\pi = 0$$ - Point condition satisfied. 4. **Conclusion:** Only option D satisfies both the slope and point conditions for the tangent line. **Final answer:** $$\boxed{f(x) = 4 \cos x \text{ and } a = \frac{3\pi}{2}}$$