Question
$\mathrm{f}(x)=3^x+2$
$( \mathbf{a} )$ Find $x$ when f$(x)=245.$
$( \mathbf{b} )$ Find $x$ when $f^-1(x)=7.$
▶️Answer/Explanation
(a) $5$
(b) $2189$
(a)
$
f(x) = 3^x + 2
$
$
3^x + 2 = 245
$
$
3^x = 243
$
Since \( 243 = 3^5 \)
$
x = 5
$
(b)
$
y = 3^x + 2
$
$
y – 2 = 3^x
$
Taking the logarithm
$
x = \log_3(y – 2)
$
$
f^{-1}(x) = \log_3(x – 2)
$
$f^{-1}(x) = 7$
$
\log_3(x – 2) = 7
$
$
x – 2 = 3^7
$
\( 3^7 = 2187 \):
$
x = 2187 + 2
$
$
x = 2189
$
Question
A curve has equation \(y=x^{3}-2x^{2}+5.\)
Find the coordinates of its two stationary points.
(………….. , …………..) and (………….., …………..)
Answer/Explanation
(0,5)
\(\left ( \frac{4}{3} ,\frac{103}{27}\right )\)